Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets

Abstract

In this study, an approximation method with an integral operational matrix based on the Muntz wavelets basis is presented to solve the variational problems of moving or fixed boundary conditions and a computational algorithm is given for the suggested approach. First, the integral operational matrix is created through the Muntz wavelets. Then, by using this integral operational matrix with Lagrange multipliers, the present approach reduces the variational problem into the system of algebraic equations. This approach is examined by some illustrative examples, and the acquired results prove that the suggested approach can solve the variational problems effectively with higher accuracy. The proposed approach yields better and comparable results with some other existing schemes given in the literature. The approximate wavelet solutions derived by the suggested approach are very identical to the corresponding exact solution.

Keywords

Muntz wavelets variational problems integral operational matrix Lagrange multipliers transversality condition

1. Introduction

The problem of determining a function which optimizes a certain functional is called variational problem (VP) (Tikhomirov, 1990). The VPs have been analyzed extensively by engineers, mathematicians, and scientists. Such types of problems appear in science, engineering, and several fields of real life such as economics, biology, solid mechanics, etc (Dym and Shames, 1973; Gelfand and Fomin, 1963). Moreover, the VPs have drawn great attention in various practical applications such as heat conduction model (Myint-U and Debnath, 2007; Schechter, 1967) and a three-dimensional eddy current field model (Jian et al., 1995). The functions that extremize functionals can be determined by using the Euler–Lagrange equation (Dym and Shames, 1973; Russak, 2002), but that equation cannot always be solved. Therefore, various direct techniques based on orthogonal functions and polynomial series have been used to solve the VPs. An overall view of these methods can be explored in the research work provided by Schechter (1967), Hwang and Shih (1983), Razzaghi et al. (2012), Mashayekhi et al. (2012), Haddai et al. (2012), Zarebnia and Birjandi (2012), Zarebnia and Sarvari (2014), Jaber (2015), Yari et al. (2017), Yari and Mirnia (2021) and Shiri and Baleanu (2019). Also, Baleanu et al. (2018) and Shiri et al. (2021) introduced the applications of an operational matrix method based on orthogonal polynomials.

Some orthogonal functions given above are dependent on the entire interval which is obviously a major drawback for some analysis work, specifically systems involving local functions that vanish outside a short time period or space. Such shortcomings can be overcome by wavelet functions. Wavelets (Daubechies, 1988; 1992) have good property for approximating functions with discontinuities or sharp changes. Over the previous few decades, there were exchanges between scientific disciplines which involve the independent development of waveforms in the engineering, physics, and geology fields that have linked to several modern wavelet applications such as human vision, image compression, prediction of earthquakes, and radar. Wavelet analysis is an effective mathematical tool which has been widely used in quantum mechanics, digital image, system analysis, numerical analysis, communication, and signal processing (Chui, 1997; Mallat, 2008). Some applications of the wavelets operational matrix method are provided by Gu and Jiang (1996), Rayal and Verma (2020a), Pandey et al. (2020) and Rayal and Verma (2020b), etc.

Today, there are a lot of works on wavelet techniques to acquire the approximate solution of VPs. For example Razzaghi and Yousefi (2000) presented a direct method based on Legendre wavelets to solve the VPs. Razzaghi and Yousefi (2001) used Legendre wavelets for the numerical analysis of some nonlinear VPs. Razzaghi and Yousefi (2002) proposed a scheme based on sine–cosine wavelets for the solution of VPs. Hsiao (2004) applied a simple Haar wavelet basis method with an integral operational matrix (IOM) to solve the VPs. Khellat and Yousefi (2006) used linear Legendre mother wavelets based on multiresolution of analysis for solving the applications in calculus of variations. Arsalani and Vali (2011) and Zhu and Wang (2013) focused on the direct scheme for handling VPs by using a Chebyshev wavelet basis. The interested reader is referred to Satyanarayan et al. (2017), Keshavarz et al. (2019) and Kheirabadi et al. (2019) and the references therein for more information.

It is necessary to find the extremum of a certain functional in many problems of applied sciences and engineering such as geometry, economics, mechanics, analysis, and so on. The VPs with different boundary conditions have gained tremendous attention because of the important role of this subject in mathematics, engineering, and applied sciences. Some applications (Dyer and McReynolds, 1970; Schechter, 1967; Weinstock, 1974) of such types of VPs appear in the heat conduction problem, brachistochrone problem, Ramsey growth model, shortest path problem, ordinary differential equation, systems of boundary value problems, etc. Due to a lot of applications of the VPs with different boundary conditions in several areas, the focus of the researchers is on the numerical solutions of the VPs with different boundary conditions. So, motivated by the above discussions, in this study, we consider the following VPs with moving or fixed boundary conditions

Min J [x (t)] = \int_{0}^{1} ζ [t, x (t), \frac{d}{d t} x (t), \frac{d^{2}}{d t^{2}} x (t), \dots, \frac{d^{n}}{d t^{n}} x (t)] d t

(1)

with initial conditions

x (0) = r_{0}, \frac{d}{d t} x (0) = r_{1}, \frac{d^{2}}{d t^{2}} x (0) = r_{2}, \dots, \frac{d^{n - 1}}{d t^{n - 1}} x (0) = r_{n - 1}

(2)

and the final conditions are either

x (1) = s_{0}, \frac{d}{d t} x (1) = s_{1}, \frac{d^{2}}{d t^{2}} x (1) = s_{2}, \dots, \frac{d^{n - 1}}{d t^{n - 1}} x (1) = s_{n - 1}

(3)

x (1), \frac{d}{d t} x (1), \frac{d^{2}}{d t^{2}} x (1), \dots, \frac{d^{n - 1}}{d t^{n - 1}} x (1) unspecified

(4)

where

(d^{n - 1} / d t^{n - 1}) x (t)

denotes the (n−1)th derivative of x(t) for

n \in ℕ

In this study, the main aim is to extremize the VPs defined in equation (1) with boundary conditions given in equations (2)–(4). In the current work, an efficient approximation approach based on Muntz wavelets is introduced for solving this kind of the VPs. This approach is a common approach, but implementation with Muntz wavelets basis functions is new. The method consists of reducing the given VPs into an algebraic system by expressing the higher derivative term in the form of Muntz wavelets with the unknown wavelet coefficients. The properties of Muntz wavelets along with the IOM and the Lagrange multipliers are applied to calculate the unknown wavelet coefficients and find the approximate wavelet solution in the given problems.

The study is structured as follows: the next Section 2 contains the description of Muntz wavelets. In Section 3, the IOM is derived by using Muntz wavelets basis functions. In Section 4, a numerical approach based on Muntz wavelets for solving the problem defined in equation (1) is proposed. In Section 5, we give the convergence and error bound of the suggested scheme. Section 6 presents some examples illustrating the accuracy and efficiency of the current approach through comparison tables and graphical representations. Finally, the conclusion of whole work is made in the last section.

2. Muntz wavelets

In this section, we describe modified form of Muntz wavelets that are used in the next section. The Muntz wavelets $ψ_{n, m} (t) = ψ (k, \hat{n}, m, γ, t)$ have the following arguments: $k \in ℤ^{+}$ , $\hat{n} = n - 1$ , γ ∈ (0, 1); t is the normalized time and m is the degree of Muntz polynomials. The definition of Muntz wavelets on the interval $[0,1)$ for a given 0 < γ < 1 is as follows (Bahmanpour et al., 2018, 2019)

ψ_{n, m} (t) = {\begin{array}{l} \sqrt{(m γ + \frac{1}{2})} 2^{k / 2} P_{m} (2^{k - 1} t - \hat{n}, γ), \frac{n - 1}{2^{k - 1}} \leq t < \frac{n}{2^{k - 1}} \\ 0, otherwise \end{array}

(5)

where m = 0, 1, 2, …, (M − 1),

M \in ℤ^{+}

and n = 1, 2, …, 2^k−1. The dilation parameter is 2^−(k−1), the translation parameter is (n−1) 2^−(k−1), and the coefficient

\sqrt{(m γ + (1 / 2))}

is for orthonormality. Here, P_m(t) are the Muntz polynomials of order m. These polynomials are orthogonal under the weight function w(t) = 1 on the interval

[0,1)

and satisfying the following recursive relation (Bahmanpour et al., 2019)

P_{0} (t, γ) = 1

(6)

P_{1} (t, γ) = (1 + \frac{1}{γ}) t^{γ} - \frac{1}{γ}

(7)

P_{m + 1} (t, γ) = \frac{1}{a_{m}} [b_{m} (t) P_{m} (t, γ) - c_{m} P_{m - 1} (t, γ)]

(8)

The terms a_m, b_m(t), and c_m are calculated by the following relation as

a_{m} = a_{m}^{(0, (1 / γ) - 1)}; b_{m} (t) = b_{m}^{(0, (1 / γ) - 1)} (2 t^{γ} - 1); c_{m} = c_{m}^{(0, (1 / γ) - 1)}

(9)

where

a_{m}^{(μ, ν)} = 2 (m + 1) (m + μ + ν + 1) (2 m + μ + ν)

(10)

b_{m}^{(μ, ν)} (t) = (2 m + μ + ν + 1) [(2 m + μ + ν) (2 m + μ + ν + 2) t + μ^{2} - ν^{2}]

(11)

c_{m}^{(μ, ν)} = 2 (m + μ) (m + ν) (2 m + μ + ν + 2)

(12)

The Muntz wavelets and modified form of Muntz wavelets are identical to each other for smaller values of m. For more details, see Bahmanpour et al. (2018, 2019).

The Muntz wavelets ψ_n,m(t) given in equation (5) are orthonormal under the weight function w(t) = 1 in L²[0, 1), that is

{〈 ψ_{n, m} (t), ψ_{n^{'}, m^{'}} (t) 〉}_{w (t)} = \int_{0}^{1} ψ_{n, m} (t) ψ_{n^{'}, m^{'}} (t) w (t) d t = {\begin{array}{l} 1, for (n, m) = (n^{'}, m^{'}) \\ 0, for (n, m) \neq (n^{'}, m^{'}) \end{array}

The next subsection explains the approximation of function in the space $L^{2} [0,1)$ by the truncated series of Muntz wavelets.

2.1. Function approximation

A function h(t) defined over $[0,1)$ may be expanded by the Muntz wavelets series as

h (t) \approx \sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} ε_{n, m} ψ_{n, m} (t)

(13)

where the coefficients ɛ_n,m are given by

ε_{n, m} = {〈 h (t), ψ_{n, m} 〉}_{w (t)}

(14)

Here, the notation ${〈 ., . 〉}_{w (t)}$ denotes the inner product in $L^{2} [0,1)$ . When the series given in equation (13) are truncated, then it can be written as

h (t) \approx \sum_{n = 1}^{2^{k - 1}} \sum_{m = 0}^{M - 1} ε_{n, m} ψ_{n, m} (t) = ℰ^{T} Ψ_{\hat{m}} (t) = Ψ_{\hat{m}}^{T} (t) ℰ

(15)

where T represents the usual transposition, and

ℰ

and

Ψ_{\hat{m}} (t)

are

\hat{m} \times 1

order matrices given by

\begin{array}{l} ℰ^{T} = [ε_{1,0}, ε_{1,1}, \dots, ε_{1, M - 1}, ε_{2,0}, ε_{2,1}, \dots, ε_{2, M - 1}, \dots, ε_{2^{k - 1}, 0}, \\ ε_{2^{k - 1}, 1}, \dots, ε_{2^{k - 1}, M - 1}] \\ = [ε_{1}, ε_{2}, \dots, ε_{\hat{m}}] \end{array}

(16)

\begin{array}{l} Ψ_{\hat{m}}^{T} (t) = [ψ_{1,0} (t), ψ_{1,1} (t), \dots, ψ_{1, M - 1} (t), ψ_{2,0} (t), ψ_{2,1} (t), \dots, \\ ψ_{2, M - 1} (t), \dots, ψ_{2^{k - 1}, 0} (t), ψ_{2^{k - 1}, 1} (t), \dots, ψ_{2^{k - 1}, M - 1} (t)] \\ = [ψ_{1}, ψ_{2}, \dots, ψ_{\hat{m}}] \end{array}

(17)

Here, $2^{k - 1} M = \hat{m}$ is the total number of Muntz wavelets basis functions.

After this section, the IOM is constructed for the Muntz wavelets basis which can play an important role to convert the given problems into the algebraic system.

3. Constructing Muntz wavelets integral operational matrix

In this section, we construct the Muntz wavelets operational matrix $ℙ_{\hat{m} \times \hat{m}}$ of integration.

For $\hat{m} = 6 (k = 1, M = 6)$ and $γ = 1 / 2$ , we get six Muntz wavelet basis functions on [0,1) using equations (5)–(12) as

ψ_{1,0} (t) = 1

ψ_{1,1} (t) = \sqrt{2} (3 \sqrt{t} - 2)

ψ_{1,2} (t) = \frac{1}{12 \sqrt{3}} (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t})) (- 2 + 3 \sqrt{t}))

\begin{array}{l} ψ_{1,3} (t) = \frac{1}{60} [\frac{1}{6} (- 1 + 35 (- 1 + 2 \sqrt{t})) \\ \times (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t}) \\ \times (- 2 + 3 \sqrt{t})) - 84 (- 2 + 3 \sqrt{t}))] \end{array}

\begin{array}{l} ψ_{1,4} (t) = \frac{1}{56 \sqrt{5}} [\frac{1}{15} (- 1 + 63 (- 1 + 2 \sqrt{t})) \\ \times (\frac{1}{6} (- 1 + 35 (- 1 + 2 \sqrt{t})) \\ \times (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t})) \\ \times (- 2 + 3 \sqrt{t})) - 84 (- 2 + 3 \sqrt{t})) \\ \times - 6 (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t})) \times (- 2 + 3 \sqrt{t}))] \end{array}

\begin{array}{l} ψ_{1,5} (t) = \frac{1}{90 \sqrt{6}} [\frac{1}{28} (\frac{1}{15} (- 1 + 63 (- 1 + 2 \sqrt{t})) \\ \times (\frac{1}{6} (- 1 + 35 (- 1 + 2 \sqrt{t})) \\ \times (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t})) (- 2 + 3 \sqrt{t})) \\ \times - 84 (- 2 + 3 \sqrt{t})) - 6 (- 20 + 4 (- 1 + 15 \times (- 1 + 2 \sqrt{t})) \\ (- 2 + 3 \sqrt{t}))) (- 1 + 99 \times (- 1 + 2 \sqrt{t})) \\ - \frac{11}{3} (\frac{1}{6} (- 1 + 35 (- 1 + 2 \sqrt{t})) \\ \times (- 20 + 4 (- 1 + 15 (- 1 + 2 \sqrt{t})) \\ \times (- 2 + 3 \sqrt{t})) - 84 (- 2 + 3 \sqrt{t}))] \end{array}

By using equation (17), we can write the above wavelets basis as

Ψ_{6} (t) = {[ψ_{1,0} (t), ψ_{1,1} (t), ψ_{1,2} (t), ψ_{1,3} (t), ψ_{1,4} (t), ψ_{1,5} (t)]}^{T}

(18)

By integrating equation (18) with respect to t from 0 to t and expressing in matrix form, we obtain

\int_{0}^{t} Ψ_{6} (t^{'}) d t^{'} \approx ℙ_{6 \times 6} Ψ_{6} (t)

(19)

where

ℙ_{6 \times 6} = [\begin{matrix} \frac{1}{2} & \frac{\sqrt{2}}{5} & \frac{1}{10 \sqrt{3}} & 0 & 0 & 0 \\ - \frac{\sqrt{2}}{5} & 0 & \frac{\sqrt{2}}{7 \sqrt{3}} & \frac{\sqrt{2}}{35} & 0 & 0 \\ - \frac{1}{10 \sqrt{3}} & - \frac{\sqrt{2}}{7 \sqrt{3}} & 0 & \frac{2}{15 \sqrt{3}} & \frac{\sqrt{5}}{42 \sqrt{3}} & 0 \\ 0 & - \frac{\sqrt{2}}{35} & - \frac{2}{15 \sqrt{3}} & 0 & \frac{2 \sqrt{5}}{77} & \frac{\sqrt{2}}{33 \sqrt{3}} \\ 0 & 0 & - \frac{\sqrt{5}}{42 \sqrt{3}} & - \frac{2 \sqrt{5}}{77} & 0 & \frac{\sqrt{10}}{39 \sqrt{3}} \\ 0 & 0 & 0 & - \frac{\sqrt{2}}{33 \sqrt{3}} & - \frac{\sqrt{10}}{39 \sqrt{3}} & 0 \end{matrix}]

also, we derive the following result as

\int_{0}^{t} Ψ_{6} (t^{'}) d t^{'} = ℙ_{6 \times 6} Ψ_{6} (t) + {\bar{Ψ}}_{6} (t) + {\bar{\bar{Ψ}}}_{6} (t)

where

{\bar{Ψ}}_{6} (t) = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \frac{\sqrt{35}}{286} ψ_{1,6} (t) \\ 0 \end{matrix}]

{\bar{\bar{Ψ}}}_{6} (t) = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \frac{\sqrt{14}}{55 \sqrt{3}} ψ_{1,6} (t) + \frac{2}{65 \sqrt{3}} ψ_{1,7} (t) \end{matrix}]

In this study, we use the result given in equation (19) for solving VPs. In this way, we can derive the other matrix by consuming more number of basis functions.

In general, we have

\int_{0}^{t} Ψ_{\hat{m}} (t^{'}) d t^{'} \approx ℙ_{\hat{m}} Ψ_{\hat{m}} (t)

(20)

where

Ψ_{\hat{m}} (t)

given in equation (17) and

ℙ_{\hat{m} \times \hat{m}}

is the Muntz wavelets integral operational matrix (MWIOM) given by

ℙ_{\hat{m} \times \hat{m}} = \frac{1}{2^{k - 1}} [\begin{matrix} F & S & S & \dots & S \\ O & F & S & ⋱ & ⋮ \\ O & O & F & ⋱ & S \\ ⋮ & ⋱ & ⋱ & ⋱ & S \\ O & \dots & O & O & F \end{matrix}]

Here, O, S, and F are M × M order matrices given by

O = [\begin{matrix} 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 \end{matrix}]

S = [\begin{matrix} 1 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 \end{matrix}]

and

F_{i, j} = {〈 g_{i} (t), Ψ_{j} (t) 〉}_{w (t)}; i, j = 1,2, \dots, M

where

g_{i} (t) = \int_{0}^{t} Ψ_{i} (t^{'}) d t^{'}

The integration of the product of two Muntz wavelets function vectors is obtained as

ℐ = \int_{0}^{1} Ψ_{\hat{m}} (t) Ψ_{\hat{m}}^{T} (t) d t

(21)

where

ℐ

is an identity matrix. The result in equation (21) is more useful for solving VPs.

Advantages of using Muntz wavelets basis functions:

Only a few numbers of the Muntz wavelet basis functions are needed to attain very satisfactory results. This observation is given by numerical examples and by comparing it with other existing methods given in the literature.

The integration of the product of two Muntz wavelet function vectors is an identity matrix which is computationally more attractive and simpler.

The MWIOM contains many zeros that play a key role in simplification of the performance index.

In the next section, we give the details of suggested numerical technique and its algorithm for solving the given VPs.

4. Method of solution

The procedure of the proposed scheme is given stepwise as:

For the VPs given in equation (1), approximate the higher derivative function $(d^{n} / d t^{n}) x (t)$ in the form of Muntz wavelets series according to the function approximation given in subsection 2.1 as

\frac{d^{n}}{d t^{n}} x (t) \approx ℰ^{T} Ψ_{\hat{m}} (t) = Ψ_{\hat{m}}^{T} (t) ℰ = \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t)

(22)

where the truncated wavelets series approximation of

(d^{n} / d t^{n}) x (t)

is denoted by

(d^{n} / d t^{n}) x_{\hat{m}} (t)

, and the matrices

ℰ

Ψ_{\hat{m}} (t)

are given in equations (16) and (17), respectively.

2. Approximation of x(t) can be computed as follows:

a. Integrating equation (22) with respect to t from 0 to t as

\int_{0}^{t} \frac{d^{n}}{d t^{n}} x (t) d t \approx \int_{0}^{t} ℰ^{T} Ψ_{\hat{m}} (t) d t

which implies

\begin{array}{l} \frac{d^{n - 1}}{d t^{n - 1}} x (t) \approx \int_{0}^{t} ℰ^{T} Ψ_{\hat{m}} (t) d t + \frac{d^{n - 1}}{d t^{n - 1}} x (0) \\ \approx ℰ^{T} ℙ_{\hat{m} \times \hat{m}} Ψ_{\hat{m}} (t) + \frac{d^{n - 1}}{d t^{n - 1}} x (0) \end{array}

(23)

where

ℙ_{\hat{m} \times \hat{m}}

is the MWIOM given in equation (20).

b. Again, integrating equation (23) from 0 to t, we get

\begin{array}{l} \frac{d^{n - 2}}{d t^{n - 2}} x (t) \approx ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{2} Ψ_{\hat{m}} (t) + \int_{0}^{t} \frac{d^{n - 1}}{d t^{n - 1}} x (0) d t + \frac{d^{n - 2}}{d t^{n - 2}} x (0) \\ = ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{2} Ψ_{\hat{m}} (t) + t \frac{d^{n - 1}}{d t^{n - 1}} x (0) + \frac{d^{n - 2}}{d t^{n - 2}} x (0) \end{array}

(24)

c. Again, integrating equation (24) from 0 to t, we have

\begin{array}{l} \frac{d^{n - 3}}{d t^{n - 3}} x (t) \approx ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{3} Ψ_{\hat{m}} (t) + \frac{t^{2}}{2!} \frac{d^{n - 1}}{d t^{n - 1}} x (0) + t \frac{d^{n - 2}}{d t^{n - 2}} x (0) \\ + \frac{d^{n - 3}}{d t^{n - 3}} x (0) \end{array}

(25)

In this way, we can obtain x(t) by the successive integration of equation (25) as

\begin{array}{l} x (t) \approx ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{n} Ψ_{\hat{m}} (t) + \frac{t^{n - 1}}{(n - 1)!} \frac{d^{n - 1}}{d t^{n - 1}} x (0) + \dots + \frac{t^{3}}{3!} \frac{d^{3}}{d t^{3}} x (0) \\ + \frac{t^{2}}{2!} \frac{d^{2}}{d t^{2}} x (0) + t \frac{d}{d t} x (0) + x (0) \end{array}

(26)

3. Approximate each known functions tⁱ and r_n−i by Muntz wavelets as

t^{i} \approx ℒ_{i}^{T} Ψ_{\hat{m}} (t); i = 1,2, \dots, n - 1

(27)

r_{n - i} \approx ℛ_{n - i}^{T} Ψ_{\hat{m}} (t); i = 1,2, \dots, n

(28)

4. Substitute equations (2), (27), and (28) in (26), we get

\begin{array}{l} x (t) \approx [ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{n} + \frac{r_{n - 1}}{(n - 1)!} ℒ_{n - 1}^{T} + \dots + \frac{r_{3}}{3!} ℒ_{3}^{T} + \frac{r_{2}}{2!} ℒ_{2}^{T} \\ + r_{1} ℒ_{1}^{T} + ℛ_{0}^{T}] Ψ_{\hat{m}} (t) \\ = G^{T} Ψ_{\hat{m}} (t) \end{array}

(29)

where

\begin{array}{l} G^{T} = ℰ^{T} {(ℙ_{\hat{m} \times \hat{m}})}^{n} + \frac{r_{n - 1}}{(n - 1)!} ℒ_{n - 1}^{T} + \dots + \frac{r_{3}}{3!} ℒ_{3}^{T} + \frac{r_{2}}{2!} ℒ_{2}^{T} \\ + r_{1} ℒ_{1}^{T} + ℛ_{0}^{T} \end{array}

5. Substituting equations (2), (22)–(25), (27), and (29) in (1), then the VPs defined in equation (1) is transformed into the approximate form as

J [x (t)] \approx \int_{0}^{1} ζ [ℒ^{T} Ψ_{\hat{m}} (t), G^{T} Ψ_{\hat{m}} (t), \dots, ℰ^{T} Ψ_{\hat{m}} (t)] d t = J [ℰ]

(30)

6. Approximate the final boundary conditions as:

a. The boundary conditions mentioned in equation (3) can be written as

c_{j} = \frac{d^{j}}{d t^{j}} x (1) - s_{j} = 0; j = 0,1, \dots, n - 1

(31)

The value of c_j can be expressed in the form of wavelets series expansion by using equations (23)–(25) and (29).

b. The boundary conditions in equation (4) which are unspecified can be expressed by the transversality condition (Dym and Shames, 1973; Russak, 2002) as

{\frac{\partial ζ}{\partial x^{(j)} (t)} |}_{t = 1} = 0

(32)

by solving equation (32), we obtain c_j as equation (31) which can also be approximated in the term of wavelets series by using equations (23)–(25) and (29).

7. Consider the function

J^{*} [ℰ, λ] = J [ℰ] + λ C

where λ = [λ₀, λ₁, …, λ_n−1] is the unknown Lagrange multipliers and

C = {[c_{0}, c_{1}, \dots, c_{n - 1}]}^{T}

8. The necessary condition for the extremal of the VP defined in equation (1) is given as

\nabla J^{*} [ℰ, λ] = 0

(33)

where ∇ represents the partial derivative of J^* with respect to

ℰ

and λ.

9. Solve the system of algebraic equations obtained from equation (33), we can easily determine the approximate wavelets solution of x(t) which gives the optimal values of functional defined in equation (1).

Procedure completed.

The proposed scheme is algorithmically expressed as follows:

Algorithm

Input: $M, n \in ℕ$ ; $k, j \in ℕ \cup {0}$ ; $\hat{m} = 2^{k - 1} M$ ; r₀, r₁, …, r_n−1; s₀, s₁, …, s_n−1 and the function ζ.

Step 1

Choose M and k.

Step 2

Compute the Muntz wavelets $Ψ_{\hat{m}} (t)$ from equation (17).

Step 3

Compute the Muntz wavelets operational matrix $ℙ_{\hat{m} \times \hat{m}}$ of integration using equation (20).

Step 4

Define unknown vector $ℰ$ from equation (16).

Step 5

Determine the vector $G$ in equation (29) and vector $ℒ$ in equation (27).

Step 6

Compute the equation.

$J [ℰ] = \int_{0}^{1} ζ [ℒ^{T} Ψ_{\hat{m}} (t), G^{T} Ψ_{\hat{m}} (t), \dots, ℰ^{T} Ψ_{\hat{m}} (t)] d t$ by using equations (28) and (29).

Step 7

Determine c_j by equation (31) or (32).

Step 8

Put $λ = [λ_{0}, λ_{1}, \dots, λ_{n - 1}] and C = {[c_{0}, c_{1}, \dots, c_{n - 1}]}^{T}$ .

Step 9

Put $J^{*} [ℰ, λ] = J [ℰ] + λ C$ .

Step 10

Solve the set of linear algebraic equations: $\nabla J^{*} [ℰ, λ] = 0$ for the vectors $ℰ$ and λ.

Step 11

Increase M or k to achieve better approximation of x(t) and J.

Output: the approximated solution $x (t) \approx G^{T} Ψ_{\hat{m}} (t)$ and approximated performance index J^* = J.

5. Error and convergence analysis

In this section, we give the convergence and error analysis of the suggested scheme. For this, we present the following notation and theorems.

Let $x_{\hat{m}} (t)$ be the Muntz wavelets approximate solution of x(t), where x(t) is the exact solution of equations (1)–(4). Then, the absolute error function e_A(t) is calculated as

e_{A} (t) = | x (t) - x_{\hat{m}} (t) |, t \in [0,1]

The upper bound of the approximation error by using the Muntz wavelets series is given by the following theorem.

Theorem 1

Let $x^{(i)} (t) \in C [0,1]$ , i = 0, 1, 2, …M, and $x_{\hat{m}} (t) = A^{T} Ψ_{\hat{m}} (t)$ are the best approximation to x(t) from $y_{\hat{m}} = span {ψ_{1,0} (t), ψ_{1,1} (t), \dots, ψ_{1, (M - 1)} (t), ψ_{2,0} (t), ψ_{2,1} (t), \dots, ψ_{2, (M - 1)} (t), ψ_{2^{k - 1}, 0} (t), ψ_{2^{k - 1}, 1} (t), \dots, ψ_{2^{k - 1}, (M - 1)} (t)}$ , then the error bound of approximate solution is given as

{‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \leq \frac{Ω}{Γ (M + 1) \sqrt{2 M + 1}}

where

Ω \geq sup_{t \in [0,1]} | x^{(M)} (t) |

Proof

Consider the classical Taylor’s formula as (Lang, 1986; Odibat and Shawagfeh, 2007)

x (t) = \sum_{i = 0}^{M - 1} \frac{t^{i}}{Γ (i + 1)} x^{(i)} (0^{+}) + \frac{t^{M}}{Γ (M + 1)} x^{(M)} (ξ)

with

ξ \in (0, t], \forall 0 \leq t \leq 1

and

| x (t) - \sum_{i = 0}^{M - 1} \frac{t^{i}}{Γ (i + 1)} x^{(i)} (0^{+}) | \leq \frac{t^{M}}{Γ (M + 1)} sup_{t \in [0,1]} | x^{(M)} (t) |

(34)

Because $x_{\hat{m}} (t) = A^{T} Ψ_{\hat{m}} (t)$ is the best approximation to x(t) out of $y_{\hat{m}}$ and $\sum_{i = 0}^{M - 1} (t^{i} / Γ (i + 1)) x^{(i)} (0^{+}) \in y_{\hat{m}}$ , then

\begin{array}{l} {‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)}^{2} \leq {‖ x (t) - \sum_{i = 0}^{M - 1} \frac{t^{i}}{Γ (i + 1)} x^{(i)} (0^{+}) ‖}_{L^{2} [0,1)}^{2} \\ = \int_{0}^{1} {(\frac{t^{M}}{Γ (M + 1)} sup_{t \in [0,1]} | x^{(M)} (t) |)}^{2} w (t) d t \\ = {(\frac{Ω}{Γ (M + 1)})}^{2} \int_{0}^{1} t^{2 M} w (t) d t \\ = {(\frac{Ω}{Γ (M + 1)})}^{2} \frac{1}{(2 M + 1)} \end{array}

where w(t) = 1 is a weight function for Muntz wavelets, and

Ω \geq sup_{t \in [0,1]} | x^{(M)} (t) |

By taking square root on both sides of the above inequality, we obtain the desired result as

{‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \leq \frac{Ω}{Γ (M + 1) \sqrt{2 M + 1}}

Also, when M → ∞, then ${‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \to 0$ .

Theorem 2

Suppose the assumptions of Theorem 1 hold, then we have

{‖ \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \leq \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}}

Proof

According to equation (34), we have

\begin{array}{l} | \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) | \leq \frac{Ω}{Γ (M + 1)} \frac{d^{n}}{d t^{n}} (t^{M}) \\ = \frac{Ω}{Γ (M + 1)} [M (M - 1) (M - 2) (M - 3) \\ \times \dots (M - n + 1) t^{M - n}] \\ = \frac{Ω}{Γ (M - n + 1)} t^{M - n} \end{array}

then

\begin{array}{l} {‖ \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)}^{2} \leq \int_{0}^{1} {(\frac{Ω}{Γ (M - n + 1)} t^{M - n})}^{2} d t \\ = {(\frac{Ω}{Γ (M - n + 1)})}^{2} \int_{0}^{1} t^{2 M - 2 n} d t \\ = {(\frac{Ω}{Γ (M - n + 1)})}^{2} \frac{1}{(2 M - 2 n + 1)} \end{array}

By taking square root on both sides of the above inequality, we obtain the desired result as

{‖ \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \leq \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}}

Theorem 3

Suppose the assumptions of Theorem 1 and Theorem 2 hold and the function ζ in equation (1) satisfies the Lipschitz condition. Then, the error bound E_A of the suggested method is given by

{‖ E_{A} ‖}_{L^{2} [0,1)} < β (n + 1) \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}}

where

E_{A} = J [x (t)] - J [x_{\hat{m}} (t)]

, and β is a Lipschitz constant.

Proof

We have

{‖ E_{A} ‖}_{L^{2} [0,1)} = {‖ J [x (t)] - J [x_{\hat{m}} (t)] ‖}_{L^{2} [0,1)}

By using equation (1), we obtain

{‖ E_{A} ‖}_{L^{2} [0,1)} = {‖ \int_{0}^{1} ℱ d t - \int_{0}^{1} ℱ_{\hat{m}} d t ‖}_{L^{2} [0,1)}

where for brevity, we put

ℱ = ζ [t, x (t), \frac{d}{d t} x (t), \dots, \frac{d^{n}}{d t^{n}} x (t)]

and

ℱ_{\hat{m}} = ζ [t, x_{\hat{m}} (t), \frac{d}{d t} x_{\hat{m}} (t), \dots, \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t)]

Since ζ satisfies a Lipschitz condition, we get

\begin{array}{l} {‖ E_{A} ‖}_{L^{2} [0,1)} \leq β \int_{0}^{1} {‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} d t \\ + β \int_{0}^{1} {‖ \frac{d}{d t} x (t) - \frac{d}{d t} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} d t \\ + \dots + β {\int_{0}^{1} ‖ \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} d t \end{array}

where β is a Lipschitz constant. After simplification, we get

\begin{array}{l} {‖ E_{A} ‖}_{L^{2} [0,1)} \leq β {‖ x (t) - x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \\ + β {‖ \frac{d}{d t} x (t) - \frac{d}{d t} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \\ + \dots + β {‖ \frac{d^{n}}{d t^{n}} x (t) - \frac{d^{n}}{d t^{n}} x_{\hat{m}} (t) ‖}_{L^{2} [0,1)} \end{array}

(35)

By using Theorem 1 and Theorem 2 in equation (35), we obtain

\begin{array}{l} {‖ E_{A} ‖}_{L^{2} [0,1)} \leq β \frac{Ω}{Γ (M + 1) \sqrt{2 M + 1}} + β \frac{Ω}{Γ (M) \sqrt{2 M - 1}} \\ + \dots + β \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \end{array}

(36)

Because

\begin{array}{l} Γ (M + 1) \sqrt{2 M + 1} > Γ (M) \sqrt{2 M - 1} \\ > \dots > Γ (M - n + 1) \sqrt{2 M - 2 n + 1} \end{array}

then

\begin{array}{l} \frac{1}{Γ (M + 1) \sqrt{2 M + 1}} < \frac{1}{Γ (M) \sqrt{2 M - 1}} \\ < \dots < \frac{1}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \end{array}

Using the above relation in equation (36), we get

\begin{array}{l} {‖ E_{A} ‖}_{L^{2} [0,1)} < β \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \\ + β \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \\ + \dots + β \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \\ = β (n + 1) \frac{Ω}{Γ (M - n + 1) \sqrt{2 M - 2 n + 1}} \end{array}

(37)

Hence, the desired result is obtained.

Also, when M → ∞, then ${‖ E_{A} ‖}_{L^{2} [0,1)} \to 0$ .

Next, the procedure introduced in Section 4 is applied to solve the given VPs.

6. Method implementation

In this section, five examples are taken to show the efficiency of the suggested scheme. The approximate wavelets solutions calculated by the constructed scheme are compared to the solution obtained using other existing schemes and the corresponding exact solution. All computations are made by software Mathematica 7.

6.1. Example 1

Consider the following VP (Razzaghi and Yousefi, 2000)

Min J [x (t)] = \int_{0}^{1} [{\frac{d}{d t} x (t)}^{2} + t \frac{d}{d t} x (t) + {x (t)}^{2}] d t

with the boundary conditions

x (0) = 0, x (1) = 0.25

The exact solution for this problem by Euler’s equation is given as

x_{e} (t) = \frac{e^{- t} (- 1 + e^{t}) (e - 2 e^{2} - 2 e^{t} + e^{1 + t})}{4 (1 - e^{2})}

and

J_{e} = 0.19759399465871066

We solve the given problem for M = 5, 6, 8, 10 and k = 1 by using described procedure in Section 4. In Table 1, the approximate results calculated by the presented scheme are compared to the results derived by the approaches in Razzaghi and Yousefi (2000) and Razzaghi and Ordokhani (2001) along with the exact solution. In the techniques of Razzaghi and Yousefi (2000) and Razzaghi and Ordokhani (2001), the solution is derived with 12 and 8 basis functions, respectively. The obtained approximate solutions and their absolute errors are graphically presented in Figure 1. Table 2 provides the comparison in absolute errors between the proposed scheme and the method of Bokhari et al. (2018). It is clearly shown from Table 2 that the results found by the suggested method are superior to Bokhari et al. (2018). The absolute errors and the appropriate values of J for this problem by the suggested method for k = 1 and several values of M are displayed in Table 3 which describes the efficiency of the presented scheme. Also, it is clearly seen that the suggested approach and the exact solution are in perfect harmony.

Table 1.

Comparison of approximate solutions in Example 1.

t	Method in Razzaghi and Yousefi (2000)	Method in Razzaghi and Ordokhani (2001)	Present method, k = 1			Exact
t	Method in Razzaghi and Yousefi (2000)	Method in Razzaghi and Ordokhani (2001)	M = 6	M = 8	M = 10	Exact
0.0	0.000000	0.0000	0.000117269	0.000000107	0.000000108	0.000000000
0.1	0.041949	0.0396	0.041961873	0.041950733	0.041950727	0.041950728
0.2	0.079315	0.0761	0.079318996	0.079317151	0.079317148	0.079317146
0.3	0.112471	0.1146	0.112463398	0.112473206	0.112473225	0.112473228
0.4	0.141749	0.1482	0.141744509	0.141750817	0.141750814	0.141750812
0.5	0.167443	0.1817	0.167447441	0.167442943	0.167442921	0.167442918
0.6	0.189807	0.1817	0.189817055	0.189806682	0.189806678	0.189806681
0.7	0.209064	0.2078	0.209070632	0.209065895	0.209065923	0.209065925
0.8	0.225411	0.2267	0.225405195	0.225413395	0.225413406	0.225413402
0.9	0.239010	0.2398	0.239002143	0.239012764	0.239012723	0.239012725
1.0	0.249999	0.2515	0.250030378	0.249999904	0.250000012	0.250000000

Figure 1.

Left: Approximate solution for k = 1 and M = 6, Right: Absolute error functions for k = 1 and different values of M in Example 1.

Table 2.

Comparison of absolute errors in Example 1.

t	Legendre wavelets method (Bokhari et al., 2018), k = 2		Present method
	Legendre wavelets method (Bokhari et al., 2018), k = 2		k = 1	k = 1	k = 1
	M = 4	M = 6	M = 6	M = 8	M = 10
0.1	6.7 × 10⁻⁶	5.5 × 10⁻⁹	1.1 × 10⁻⁵	5.1 × 10⁻⁹	1.3 × 10⁻⁹
0.2	1.0 × 10⁻⁵	3.0 × 10⁻⁹	1.8 × 10⁻⁶	5.3 × 10⁻⁹	1.7 × 10⁻⁹
0.3	9.8 × 10⁻⁶	2.8 × 10⁻⁹	9.8 × 10⁻⁶	2.2 × 10⁻⁸	3.5 × 10⁻⁹
0.4	6.8 × 10⁻⁶	5.4 × 10⁻⁹	6.3 × 10⁻⁶	4.7 × 10⁻⁹	1.7 × 10⁻⁹
0.5	6.0 × 10⁻⁹	1.7 × 10⁻¹⁵	4.5 × 10⁻⁶	2.5 × 10⁻⁸	2.6 × 10⁻⁹
0.6	4.7 × 10⁻⁶	3.8 × 10⁻⁹	1.0 × 10⁻⁵	8.0 × 10⁻¹⁰	2.5 × 10⁻⁹
0.7	7.1 × 10⁻⁶	2.1 × 10⁻⁹	4.7 × 10⁻⁶	2.9 × 10⁻⁸	1.9 × 10⁻⁹
0.8	6.9 × 10⁻⁶	2.0 × 10⁻⁹	8.2 × 10⁻⁶	7.1 × 10⁻⁹	3.7 × 10⁻⁹
0.9	4.7 × 10⁻⁶	3.8 × 10⁻⁹	1.0 × 10⁻⁵	3.9 × 10⁻⁸	1.9 × 10⁻⁹

Table 3.

Absolute errors and the approximate values of the performance index J for Example 1.

M	Present method	Absolute error
5	0.19759400600773583	1.13 × 10⁻⁸
6	0.19759399457257665	8.61 × 10⁻¹¹
8	0.19759399465875452	4.38 × 10⁻¹⁴
10	0.19759399465871070	2.77 × 10⁻¹⁷

6.2. Example 2

Consider the following VP (Shihab and Asma, 2012)

Min J [x (t)] = \int_{0}^{1} [{\frac{d}{d t} x (t)}^{2} + {x (t)}^{2}] d t

with the boundary conditions

x (0) = 0, x (1) = 1

The exact solution for this problem by Euler’s equation is given as

x_{e} (t) = \frac{e^{t} - e^{- t}}{e^{1} - e^{- 1}}

and

J_{e} = 1.3130352854993312

We solve the above problem for k = 1 and M = 5, 6, 8, 10 by using the described procedure in Section 4. In Table 4, the approximate solution derived by proposed scheme is compared with the solutions obtained through the method of Shihab and Asma (2012) together with the exact solution. The obtained approximate solutions and their absolute errors are graphically represented in Figure 2. In Table 5, absolute errors and the optimal values of J for k = 1 and different values of M are given. Table 5 shows that very high accuracy is achieved by the presented method. From Table 4 and Figure 2, it is clearly seen that the results found by the suggested method are superior to Shihab and Asma (2012). Also, the exact solution and the presented method are in nice agreement. In addition, using more Muntz wavelet basis functions improves the accuracy rapidly.

Table 4.

Comparison of approximate solutions in Example 2.

t	Chebyshev wavelets method	Second kind Chebyshev wavelets method	Present method, k = 1			Exact
t	Chebyshev wavelets method	Second kind Chebyshev wavelets method	M = 6	M = 8	M = 10	Exact
0.0	0.000576	0.001149	0.0011434	0.0000364	0.0000006	0.0000000
0.1	0.084691	0.085051	0.0851405	0.0852346	0.0852337	0.0852337
0.2	0.170974	0.171115	0.1713136	0.1713209	0.1713204	0.1713204
0.3	0.294224	0.259339	0.2591954	0.2591201	0.2591218	0.2591218
0.4	0.350037	0.349725	0.3495596	0.3495164	0.3495166	0.3495166
0.5	0.442818	0.442767	0.4433763	0.4434110	0.4434094	0.4434094
0.6	0.541037	0.540356	0.5416680	0.5417406	0.5417400	0.5417400
0.7	0.645009	0.644021	0.6454597	0.6454911	0.6454926	0.6454926
0.8	0.756033	0.754707	0.7557600	0.7557045	0.7557054	0.7557054
0.9	0.874108	0.872416	0.8735535	0.8734837	0.8734817	0.8734817
1.0	0.999234	0.997145	0.9997983	0.9999948	1.0000000	1.0000000

Figure 2.

Left: Approximate solution for k = 1 and M = 6, Right: Absolute error functions for k = 1 and different values of M in Example 2.

Table 5.

Absolute errors and the approximate values of the performance index J for Example 2.

M	Present method	Absolute error
5	1.3130351306065302	1.54 × 10⁻⁷
6	1.3130352866748982	1.17 × 10⁻⁹
8	1.3130352854987295	6.01 × 10⁻¹³
10	1.3130352854993330	1.77 × 10⁻¹⁵

6.3. Example 3

Consider the following VP (Hsiao, 2004)

Min J [x (t)] = \int_{0}^{1} [{\frac{d}{d t} x (t)}^{2} + t \frac{d}{d t} x (t)] d t

with the boundary conditions

x (0) = 0, x (1) is unspecified

The exact solution to this problem by Euler’s equation is given as

x_{e} (t) = - \frac{t^{2}}{4}

and

J_{e} = - 0.08333333333333333

We solve this problem for M = 4, 5, 6 and k = 1 by using the described procedure in Section 4. The obtained wavelets solutions and their absolute errors are depicted in Figure 3. In Table 6, the approximate results calculated by the proposed scheme for M = 5 and k = 1 are compared with the results derived by the Haar wavelets method (HWM) (Osama et al., 2014), Hat basis solution (Osama, 2018), second kind Chebyshev wavelets method (Zhu and Wang, 2013), and sine–cosine wavelets method (SCWM) (by using 28 basis functions) together with the exact solution. In the techniques of Osama et al. (2014), Osama (2018) and Zhu and Wang (2013), the solution is obtained with 8, 9, and 12 basis functions, respectively. From Figure 3, we see that as the value of M increases, the absolute errors decrease which provide the information about the convergence of the suggested scheme. The absolute error in the optimal value of J to this problem by the suggested method for M = 5 and k = 1 is 4.16 × 10⁻¹⁷.

Figure 3.

Left: Approximate solution for k = 1 and M = 5, Right: Absolute error functions for k = 1 and different values of M in Example 3.

Table 6.

Comparison of approximate solutions in Example 3.

t	HWM (Osama et al., 2014)	HBS (Osama, 2018)	SKCWM (Zhu and Wang, 2013)	SCWM (Zhu and Wang, 2013)	Present method	Exact
0.000	0.001	0.0000	0.0000	−0.0078	0.0000000	0.0000000
0.125	−0.008	−0.0039	−0.0039	−0.0043	−0.0039062	−0.0039062
0.250	−0.029	−0.0156	−0.0156	−0.0391	−0.0156249	−0.0156249
0.375	−0.049	−0.0352	−0.0351	−0.0355	−0.0351562	−0.0351562
0.500	−0.108	−0.0625	−0.0625	−0.1016	−0.0624999	−0.0625000
0.625	−0.120	−0.0977	−0.0977	−0.0980	−0.0976562	−0.0976562
0.750	−0.186	−0.1406	−0.1406	−0.1964	−0.1406250	−0.1406250
0.875	−0.222	−0.1914	−0.1914	−0.1929	−0.1914062	−0.1914062
1.000	−0.222	−0.2539	−0.2500	−0.2522	−0.2500000	−0.2500000

SCWM: sine–cosine wavelets method; SKCWM: second kind Chebyshev wavelets method; HBS: Hat basis solution; HWM: Haar wavelets method.

6.4. Example 4

Consider the following VP (Arsalani and Vali, 2011)

Min J [x (t)] = \int_{0}^{1} [\frac{1}{2} {\frac{d^{2}}{d t^{2}} x (t)}^{2} + (4 - 4 t) \frac{d}{d t} x (t)] d t

under the boundary conditions

x (0) = 0, \frac{d}{d t} x (0) = 1

x(1) and $(d / d t) x (1)$ are unspecified.

The exact solution to this problem by Euler’s equation is given as

x_{e} (t) = - \frac{1}{6} t^{4} + \frac{2}{3} t^{3} - t^{2}

and

J_{e} = - 0.4000000000000007

We solve the given problem for M = 6, 7, 8 and k = 1 by using the proposed method. The obtained wavelets approximate solutions and their absolute errors are plotted in Figure 4. In Table 7, the numerical results calculated by the proposed scheme are compared with the results derived by the SCWM (by using 20 basis functions), HWM (Hsiao, 2004), and Chebyshev wavelets method (Arsalani and Vali, 2011) together with the exact solution. In the methods of Hsiao (2004) and Arsalani and Vali (2011), the solution is obtained with 8 and 6 basis functions, respectively. From Table 7, it can be seen that the convergence of the approximate wavelets solution to the exact solution as the value of M increases. The absolute error in the optimal value of J to this problem by the proposed method for M = 6 and k = 1 is 1.66 × 10⁻¹⁶.

Figure 4.

Left: Approximate solution for k = 1 and M = 6, Right: Absolute error functions for k = 1 and different values of M in Example 4.

Table 7.

Comparison of approximate solutions in Example 4.

t	SCWM (Arsalani and Vali, 2011)	HWM (Hsiao, 2004)	CWM (Arsalani and Vali, 2011)	Present method, k = 1			Exact
t	SCWM (Arsalani and Vali, 2011)	HWM (Hsiao, 2004)	CWM (Arsalani and Vali, 2011)	M = 6	M = 7	M = 8	Exact
0.000	−0.0253	−0.0000	0.0020	0.002226	−0.000815	−0.0000617	−0.0000000
0.125	−0.0155	−0.0138	−0.0161	−0.014147	−0.014365	−0.0143666	−0.0143636
0.250	−0.0119	−0.0518	−0.0523	−0.052834	−0.052702	−0.0527325	−0.0527343
0.375	−0.1069	−0.1075	−0.1066	−0.108917	−0.108799	−0.1087639	−0.1087646
0.500	−0.2513	−0.1758	−0.1765	−0.177011	−0.177104	−0.1770856	−0.1770833
0.625	−0.2494	−0.2521	−0.2537	−0.253116	−0.253262	−0.2532956	−0.2532958
0.750	−0.4119	−0.3330	−0.3335	−0.334003	−0.333964	−0.3339821	−0.3339843
0.875	−0.4116	−0.4161	−0.4158	−0.416926	−0.416754	−0.4167094	−0.4167073
1.000	−0.4946	−0.4999	−0.5001	−0.499477	−0.499883	−0.4999931	−0.5000000

SCWM: sine–cosine wavelets method; HWM: Haar wavelets method; CWM: Chebyshev wavelets method.

6.5. Example 5

Consider the following VP (Keshavarz et al., 2019)

Min J [x (t)] = \int_{0}^{1} [{\frac{d}{d t} x (t)}^{2} + 2 x (t) \frac{d}{d t} x (t) + 4 x^{2} (t)] d t

with the boundary conditions

x (0) = 1, x (1) = Free

The exact solution for this problem by Euler’s equation is given as

x_{e} (t) = \frac{3 e^{4 - 2 t} + e^{2 t}}{3 e^{4} + 1}

and

J_{e} = 0.9757273379195325

We solve this problem for M = 6, 7, 8 and k = 1 by using the proposed scheme. The obtained wavelets solutions and their absolute errors are plotted in Figure 5. In Table 8, the numerical results calculated by the presented scheme are compared with the results derived by the Legendre wavelets method (LWM) and Bernoulli wavelets method (BWM) (Keshavarz et al., 2019) together with the exact solution. From Table 8, it can be seen the convergence of the approximate solution to the exact solution as M increases. The comparison of the absolute error achieved by the present method, LWM, and BWM are given in Table 9. The absolute error in the optimal value of the performance index J to this problem for M = 6 and k = 1 by the proposed scheme is 8.77 × 10⁻¹⁰.

Figure 5.

Left: Approximate solution for k = 1 and M = 7, Right: Absolute error functions for k = 1 and different values of M in Example 5.

Table 8.

Comparison of approximate solutions in Example 5.

t	Legendre wavelets method	Bernoulli wavelets method	Present method		Exact
t	Legendre wavelets method	Bernoulli wavelets method	$\hat{m} = 7$	$\hat{m} = 8$	Exact
0.0	1.016400	1.000120	1.000487	1.000113	1.000000
0.1	0.828694	0.821349	0.821184	0.821170	0.821174
0.2	0.676765	0.675338	0.675285	0.675305	0.675305
0.3	0.557039	0.556552	0.556543	0.556543	0.556538
0.4	0.464307	0.460207	0.460123	0.460106	0.460107
0.5	0.388542	0.382194	0.382145	0.382138	0.382142
0.6	0.322464	0.319638	0.319502	0.319513	0.319514
0.7	0.270216	0.269741	0.269698	0.269713	0.269708
0.8	0.231592	0.230880	0.230733	0.230728	0.230727
0.9	0.202846	0.201029	0.201020	0.201000	0.201006
1.0	0.183732	0.179535	0.179317	0.179367	0.179352

Table 9.

Comparison of absolute errors in Example 5.

t	Legendre wavelets method k = 2, M = 4	Bernoulli wavelets method k = 2, M = 4	Present method, k = 1
t	Legendre wavelets method k = 2, M = 4	Bernoulli wavelets method k = 2, M = 4	M = 7	M = 8
0.0	1.64 × 10⁻²	3.27 × 10⁻⁴	4.87 × 10⁻⁴	1.13 × 10⁻⁴
0.1	7.52 × 10⁻³	1.75 × 10⁻⁴	9.91 × 10⁻⁶	3.91 × 10⁻⁶
0.2	1.46 × 10⁻³	3.25 × 10⁻⁵	1.96 × 10⁻⁵	4.52 × 10⁻⁸
0.3	5.01 × 10⁻⁴	1.42 × 10⁻⁵	5.15 × 10⁻⁶	4.65 × 10⁻⁶
0.4	4.20 × 10⁻³	1.00 × 10⁻⁴	1.53 × 10⁻⁵	1.32 × 10⁻⁶
0.5	6.40 × 10⁻³	5.15 × 10⁻⁵	2.69 × 10⁻⁶	4.54 × 10⁻⁶
0.6	2.95 × 10⁻³	1.24 × 10⁻⁴	1.17 × 10⁻⁵	8.79 × 10⁻⁸
0.7	5.08 × 10⁻⁴	3.24 × 10⁻⁵	1.04 × 10⁻⁵	4.74 × 10⁻⁶
0.8	8.65 × 10⁻⁴	1.53 × 10⁻⁴	6.07 × 10⁻⁶	1.22 × 10⁻⁶
0.9	1.84 × 10⁻³	2.33 × 10⁻⁵	1.39 × 10⁻⁵	5.99 × 10⁻⁶
1.0	4.38 × 10⁻³	1.83 × 10⁻⁴	3.47 × 10⁻⁵	1.45 × 10⁻⁵

7. Conclusion

This study presented a numerical scheme for solving VPs of fixed or moving boundary conditions by combining the MWIOM and the Lagrangian multipliers. This scheme has been examined for some examples, and the obtained approximate wavelet solutions for these examples show that this approach gives an accurate approximation of the exact solution. From the results, it is clearly observed that the Muntz wavelets is an appropriate mathematical tool to solve the given VPs arising in applied science and engineering. The biggest advantage of the current scheme is that the accuracy of approximate solutions and the least values of error are achieved by using only few terms of the Muntz wavelet basis functions in comparison to the method given by Hsiao (2004), Arsalani and Vali (2011), Zhu and Wang (2013), Keshavarz et al. (2019), Razzaghi and Yousefi (2000), Razzaghi and Ordokhani (2001), Bokhari et al. (2018), Osama et al. (2014) and Osama (2018). The optimal value of the performance index J with high precision is also obtained readily in all examples for smaller values of $\hat{m}$ . The proposed scheme is very convenient and easy in implementation.

Footnotes

Acknowledgements

The authors are very grateful to the editor and the anonymous referees for their careful reading, insightful comments, and helpful suggestions which have led to the improvement of the study.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Sag R Verma

References

Arsalani

Vali

(2011) Numerical solution of nonlinear variational problems with moving boundary conditions by using Chebyshev wavelets. Applied Mathematical Sciences 5(20): 947–964.

Bahmanpour

Kajani

Maleki

(2018) A Muntz wavelets collocation method for solving fractional differential equations. Computational and Applied Mathematics 37: 5514–5526.

Bahmanpour

Kajani

Maleki

(2019) Solving Fredholm integral equations of the first kind using Muntz wavelets. Applied Numerical Mathematics 143: 159–171.

Baleanu

Shiri

Srivastava

, et al. (2018) A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel. Advances in Difference Equations 353. DOI:10.1186/s13662-018-1822-5.

Bokhari

Amir

Bahri

(2018) A numerical approach to solve quadratic calculus of variation problems. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms 25: 427–440.

Chui

(1997) Wavelets: A Mathematical Tool For Signal Analysis. Philadelphia: SIAM, Vol. 1.

Daubechies

(1988) Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics 41: 909–996.

Daubechies

(1992) Ten Lectures on Wavelets. Philadelphia: SIAM.

Dyer

McReynolds

(1970) The Computation and Theory of Optimal Control. New York: Academic Press.

10.

Dym

Shames

(1973) Solid Mechanics: A Variational Approach. New York, NY: McGraw-Hill.

11.

Gelfand

Fomin

(1963) Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall.

12.

Jiang

(1996) The Haar wavelets operational matrix of integration. International Journal of Systems Science 27: 623–628.

13.

Haddai

Ordokhani

Razzaghi

(2012) Optimal control of delay systems by using a hybrid functions approximation. Journal of Optimization Theory and Applications 153: 338–356.

14.

Hsiao

C-H

(2004) Haar wavelet direct method for solving variational problems. Mathematics and Computers in Simulation 64: 569–585.

15.

Hwang

Shih

(1983) Laguerre series direct method for variational problems. Journal of Optimization Theory and Applications 39: 143–149.

16.

Jaber

(2015) An efficient algorithm for solving variational problems using hermite polynomials. Journal of Al Rafidain University College 35: 311–327.

17.

Jian

Bangding

Jihui

, et al. (1995) The new variational problem in three-dimensional eddy-current field computation. Communications in Numerical Methods in Engineering 11: 727–733.

18.

Keshavarz

Ordokhani

Razzaghi

(2019) The Bernoulli wavelets operational matrix of integration and its applications for the solution of linear and nonlinear problems in calculus of variations. Applied Mathematics and Computation 351: 83–98.

19.

Kheirabadi

Vaziri

Effati

(2019) Solving optimal control problem using Hermite wavelet. Numerical Algebra and Optimization 9(1): 101–112.

20.

Khellat

Yousefi

(2006) The linear Legendre mother wavelets operational matrix of integration and its application. Journal of the Franklin Institute 343: 181–190.

21.

Lang

(1986) Taylor’s formula. In: A First Course in Calculus. Undergraduate Texts in Mathematics: New York, NY: Springer.

22.

Mallat

(2008) A Wavelet Tour of Signal Processing: The Sparse Way. 3rd edition. Burlington: Academic Press.

23.

Mashayekhi

Ordokhani

Razzaghi

(2012) Hybrid functions approach for nonlinear constrained optimal control problems. Communications in Nonlinear Science and Numerical Simulation 17: 1831–1843.

24.

Myint-U

Debnath

(2007) Linear Partial Differential Equations for Scientists and Engineers. 4th edition. Boston: Birkhauser.

25.

Odibat

Shawagfeh

(2007) Generalized taylor's formula. Applied Mathematics and Computation 186(1): 286–293.

26.

Osama

Fadhel

Zaid

(2014) Numerical solution of fractional variational problems using direct Haar wavelet method. International Journal of Innovative Science Engineering and Technology 3(5): 12742–12750.

27.

Osama

(2018) A direct method for solving fractional order variational problems by hat basis functions. Ain Shams Engineering Journal 9: 1513–1518.

28.

Pandey

Dixit

Verma

(2020) An efficient solution of system of generalized Abel integral equations using Bernstein polynomials wavelet bases. Mathematical Sciences 14: 279–291. DOI:10.1007/s40096-020-00342-9.

29.

Rayal

Verma

(2020a) An approximate wavelets solution to the class of variational problems with fractional order. Journal of Applied Mathematics and Computing. DOI: 10.1007/s12190-020-01413-9

30.

Rayal

Verma

(2020b) Numerical Analysis of Pantograph Differential Equation of the Stretched Type Associated with Fractal-Fractional Derivatives via Fractional Order Legendre Wavelets. Chaos, Solitons Fractals 139: 110076. DOI: 10.1016/j.chaos.2020.110076

31.

Razzaghi

Ordokhani

(2001) An application of rationalized haar functions for variational problems. Applied Mathematics and Computation 122: 353–364.

32.

Razzaghi

Ordokhani

Haddadi

(2012) Direct method for variational problems by using hybrid of block-pulse and Bernoulli polynomials. Romanian Journal of Mathematics and Computer Science 2: 1–17.

33.

Razzaghi

Yousefi

(2000) Legendre wavelets direct method for variational problems. Mathematics and Computers in Simulation 53: 185–192.

34.

Razzaghi

Yousefi

(2001) Legendre wavelets method for the solution of nonlinear problems in the calculus of variations. Mathematical and Computer Modelling 34: 45–54.

35.

Razzaghi

Yousefi

(2002) Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations. International Journal of Systems Science 33: 805–810.

36.

Russak

(2002) Calculus of Variations, Ma 4311 Lecture Notes. Monterey, California: Naval Postgraduate School.

37.

Satyanarayan

Pragathi

Abdulelah

(2017) Laguerre wavelet and its programming. International Journal of Mathematics Trends and Technology (IJMTT) 49(2): 129–137.

38.

Schechter

(1967) The Variational Method in Engineering. New York: McGraw-Hill.

39.

Shihab

Asma

(2012) Some Approximate Algoithms for Variational Problems. Lambert Academic Publishing.

40.

Shiri

Baleanu

(2019) System of fractional differential algebraic equations with applications. Chaos, Solitons & Fractals 120: 203–212.

41.

Shiri

Perfilieva

Alijani

(2021) Classical approximation for fuzzy Fredholm integral equation. Fuzzy Sets Syst 404: 159–177. DOI:10.1016/j.fss.2020.03.023.

42.

Tikhomirov

(1990) Stories About Maxima and Minima. Providence, RI: American Mathematica Society.

43.

Weinstock

(1974) Calculus of Variations: With Applications to Physics and Engineering. Mineola: Dover Publications.

44.

Yari

Mirnia

(2021) Direct method for solution variational problems by using Hermite polynomials. Bol. Soc. Paran. Mat. 39(6): 223–237.

45.

Yari

Mirnia

Lakestani

, et al. (2017) Solution of optimal control problems with payoff term and fixet state endpoint by using Bezier polynomials. Boletim da Sociedade Paranaense de Matematica 35(1): 247–267.

46.

Zarebnia

Birjandi

(2012) The numerical solution of problems in calculus of variation using B-spline collocation method. Journal of Applied Mathematics 2012: 1. DOI:10.1155/2012/605741. Article ID 605741

47.

Zarebnia

Sarvari

(2014) Numerical solution of variational problems via parametric quintic spline method. Journal of Hyperstructures 3(1): 40–52.

48.

Zhu

Wang

(2013) Second Chebyshev wavelet operational matrix of integration and its application in the calculus of variations. International Journal of Computer Mathematics 90: 2338–2352.