A more efficient differential transform method for solving relativistic equation

Abstract

The differential transform method (DTM) has been presented for solving relativistic equation. The proposed method is easier and more reliable than usual DTM and other widely used techniques. The algebraic equations related to unknown coefficients of the proposed solution are linear, which simplifies the determination of them. The obtained results have been compared to those obtained by the numerical method (RK fourth order), the differential transform method, the modified Lindstedt–Poincare method and the harmonic balance method. The comparisons demonstrate that the proposed solution is closer to the numerical solution than the other methods.

Keywords

Relativistic equation differential transform method modified lindstedt–poincare method harmonic balance method power series

Introduction

The study of nonlinear problems is essential across physics, engineering, and related disciplines because many fundamental laws in these fields are expressed as differential equations that are inherently nonlinear. The mathematical exploration of nonlinear oscillatory issues is an extensive area of research. These issues are particularly difficult to control because the nonlinear characteristics of a system can change drastically with even minor alterations in key parameters, including time. The solutions of nonlinear differential equations are more challenging, as only a few of these equations have exact solutions. As a result, the development of approximate solution methods for nonlinear problems has been an active area of research for many years. Several analytical methods have been developed to determine approximate solutions for nonlinear oscillators. The relativistic equation represents a strongly nonlinear second-order equation that arises in the field of modern sciences, including mathematics, electric circuitry, particle physics, nuclear physics, aerodynamics, quantum field theory, etc. Moreover, it is important in the study of the oscillations of gravitational waves, quantum state changes, and astrophysical processes, including the dynamics of objects near black holes. Due to its strong nonlinear behavior, it is often difficult to achieve exact analytical solutions to the relativistic equation. Therefore, significant research has been channeled into the development of analytical methods that can provide accurate, effective and practicable solutions.

Many analytical methods have been introduced for solving the relativistic equation, including perturbation methods (PM),^1–8 the harmonic balance method (HBM),^9–14 the variational iteration method (VIM),^15,16 the energy balance method (EBM),^17–19 etc. As much as these methods can be used to determine rough periodic solutions, their accuracy and convergence rate often become worse when used on highly nonlinear systems. Moreover, higher-order approximations are often subject to numerous and complicated manipulations of algebraic terms. Belendez²⁰ employed the harmonic balance method (HBM) with a transformed formulation to solve the relativistic equation; however, the resulting algebraic equations are nonlinear, making their solution highly challenging. The modified Lindstedt–Poincare method⁵ has been used by Yeasmin²¹ to solve the relativistic equation directly without any transformation and has performed well for small amplitudes; however, at large amplitudes, it has produced a triangular waveform that deviates from the numerical solution.

The Differential Transform Method (DTM)^22–25 is a semi analytic method, in which the solution is expressed in polynomial-type power series expansion for solving nonlinear differential equations. The DTM provides satisfactory results within small time intervals but diverges for large time intervals. To overcome the limitation of the DTM, several authors^26–28 have modified the DTM. Though the modified differential transform method (MDTM) provides better results than the DTM, utilization of the MDTM method is very laborious. Recently, Alam et al.²⁹ have derived an analytic technique similar to the DTM for solving strong nonlinear problems. The method is independent of harmonic function and determines the related unknown coefficients from a system of linear algebraic equations involving unknown parameters. The higher-order approximate solution via this method is time-consuming, due to the arising in algebraic complexity.

In this article, a more efficient DTM is presented to solve relativistic equation. The process of formulating and determining the solution by using this method is relatively simple and straightforward. This method quickly converges to exact solutions and extensive good comparisons with the results of numerical evaluation and other existing methods.

Derivation of the relativistic equation

Consider the relativistic motion of a particle of rest mass $m$ under a one-dimensional harmonic oscillator force, $F = - k \bar{x}$ , where $k$ represents the elastic constant and $\bar{x}$ is the displacement (dimensional variable). The corresponding Newtonian equation of motion can be expressed as

F = \frac{d p}{d E},

(1)

where

\bar{t}

is the coordinate time (dimensional variable) and

p

is the relativistic momentum which can be written as follows

p = \frac{m v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}},

(2)

where

v = \frac{d \bar{x}}{d \bar{t}}

is the speed of the particle and

c

is the speed of light. Substituting equation (2) into equation (1) we obtain

F = \frac{d}{d \bar{t}} (\frac{m v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}) = \frac{m}{{(1 - \frac{v^{2}}{c^{2}})}^{\frac{3}{2}}} \frac{d v}{d \bar{t}} = \frac{m}{{[1 - (\frac{1}{c^{2}}) {(\frac{d \bar{x}}{d \bar{t}})}^{2}]}^{\frac{3}{2}}} \frac{d^{2} \bar{x}}{d {\bar{t}}^{2}} .

(3)

Substitute equation (3) into Newtonian equation of motion in the form

\frac{d p}{d \bar{t}} + k \bar{x} = 0 .

(4)

Now, substituting equation (2) into equation (4) and simplifying, it becomes

\frac{d^{2} \bar{x}}{d {\bar{t}}^{2}} + \frac{k}{m} {[1 - (\frac{1}{c^{2}}) {(\frac{d \bar{x}}{d \bar{t}})}^{2}]}^{\frac{3}{2}} \bar{x} = 0 .

(5)

From equation (5), the governing non-dimensional nonlinear differential equation of motion for the relativistic equation is as follows

\frac{d^{2} x}{d t^{2}} + {[1 - {(\frac{d x}{d t})}^{2}]}^{\frac{3}{2}} x = 0; x (0) = 0, \frac{d x}{d t} (0) = b^{'},

(6)

where

x

and

t

are dimensionless variables defined as follows

x = \frac{ω_{0} \bar{x}}{c} and t = ω_{0} \bar{t},

(7)

where

ω_{0} = \sqrt{\frac{k}{m}}

is the angular frequency for the nonrelativistic oscillator (linear oscillator).

Transformation of the non-dimensional relativistic equation²⁰

According to Belendez,²⁰ equation (6) can be written as the following system

\frac{d x}{d t} = y, \frac{d y}{d t} = - {(1 - y^{2})}^{\frac{3}{2}} x .

(8)

Hence, the system can be expressed as follows

\frac{d y}{d x} = - \frac{{(1 - y^{2})}^{\frac{3}{2}} x}{y} .

(9)

Here, $- \infty < x < + \infty$ and $- 1 < y < + 1$ . The transformation $y \to u$ is applied such that the transformation satisfies $- \infty < u < + \infty$ .

From the article,^30,31 the required transformation is

y = \frac{u}{\sqrt{1 + u^{2}}} .

(10)

Following the Belendez²⁰ technique, the associated second-order differential equation for $u$ can be formulated as follows

\ddot{u} + \frac{u}{\sqrt{1 + u^{2}}} = 0 .

(11)

Taking a new variable, $τ = ω t$ , equation (11) yields

ω^{2} u^{″} + \frac{u}{\sqrt{1 + u^{2}}} = 0, u (0) = b and u^{'} (0) = 0,

(12)

where the over dots denote differentiation with respect to

τ

Methodology

The classical differential transform method (DTM)²²

If a function $x (t)$ is analytic within a domain $G \subseteq C$ , a Taylor series expansion exists about any arbitrary point $t_{0} \in G$ . The differential transform (DT) of $x (t)$ can be expressed by the following expression

X (n) = \frac{1}{n!} {[\frac{d^{n} x (t)}{{d t}^{n}}]}_{t = 0},

(13)

In equation (13),

x (t)

is the original function and

X (n)

is the transformed function.

Then the inverse transform of $X (n)$ is defined as follows

x (t) = \sum_{n = 0}^{\infty} X (n) t^{n} .

(14)

Substituting equation (13) into equation (14), it becomes

x (t) = \sum_{n = 0}^{\infty} \frac{t^{n}}{n!} {[\frac{d^{n} x (t)}{{d t}^{n}}]}_{t = 0} .

(15)

In practical applications, the function

x (t)

is represented by a finite number of terms; consequently, equation (15) can be reformulated as follows

x (t) = \sum_{n = 0}^{Ν} X (n) t^{n},

(16)

or, x (t) = X (0) + X (1) t + X (2) t^{2} + X (3) t^{3} + \dots .

(17)

Although the coefficients $X (0)$ , $X (1)$ , and $X (2)$ , $\dots$ can be determined from the initial conditions, the derivation of a general recurrence relation becomes challenging due to the strong nonlinearity of the governing differential equation. This method provides accurate results over a very small region; however, it has a very slow convergence rate in a wider region. Moreover, this method diverges for strong nonlinear problems.

Alam et al.²⁹ Differential transform method

Consider the relativistic oscillator with the following initial conditions

\ddot{x} + {(1 - {\dot{x}}^{2})}^{\frac{3}{2}} x = 0, x (0) = 0, \dot{x} (0) = b^{*} .

(18)

According to transformation $τ = ω t$ , equation (18) readily becomes

ω^{2} x^{″} + {(1 - ω^{2} {x^{'}}^{2})}^{\frac{3}{2}} x = 0; x (0) = 0, x^{'} (0) = \frac{b^{*}}{ω} = β .

(19)

Consider a polynomial-type solution of equation (19) in the form

x (τ) = {B_{1}}^{'} φ (τ) + {B_{2}}^{'} φ^{2} (τ) + {B_{3}}^{'} φ^{3} (τ) + \dots,

(20)

where

φ (τ) = τ (π - τ)

Substituting equation (20) into equation (19) and then equating the coefficients of $φ^{0}$ , $φ^{1}$ , $φ^{2}, φ^{3}, φ^{5}, \dots$ the following algebraic equations are found

{B_{1}}^{'} - π^{2} {B_{2}}^{'} = 0,

(21)

{B_{1}}^{'} {(1 - {B_{1}}^{'}^{2} π^{2} ω^{2})}^{\frac{3}{2}} + ω^{2} (- 12 {B_{2}}^{'} + 6 π^{2} {B_{3}}^{'}) = 0,

(22)

\sqrt{1 - {B_{1}}^{'}^{2} π^{2} ω^{2}} ({B_{2}}^{'} + 6 {B_{1}}^{'}^{3} ω^{2}) + ω^{2} (- 30 {B_{3}}^{'} + 12 π^{2} {B_{4}}^{'}) = 0, \dots

(23)

Solving the above algebraic equations given in equations (21)-(23) yields the coefficients $B_{1}$ , $B_{2}$ , $\dots$ . The value of $φ (τ)$ as well as $x (τ)$ has a maximum value at $τ = π / 2$ . Hence, the amplitude of oscillation, $a$ , can be found from

a = \frac{{B_{1}}^{'} π^{2}}{4} + \frac{{B_{2}}^{'} π^{4}}{16} + \frac{{B_{3}}^{'} π^{6}}{64} + \dots .

(24)

Substituting the values of unknown coefficients (up to $B_{10}$ ) into equation (24), it becomes

a = \frac{β π}{2} - \frac{β π^{3} {(1 - β^{2} ω^{2})}^{3 / 2}}{48 ω^{2}} - \frac{β π^{5} {(- 1 + β^{2} ω^{2})}^{2} (- 1 + 10 β^{2} ω^{2})}{3840 ω^{4}} - \dots .

(25)

For this oscillator, the relation between

a

and

β

β^{2} ω^{2} = 1 - \frac{4}{{(2 + a^{2})}^{2}} .

(26)

By substituting

\frac{4}{{(2 + a^{2})}^{2}} = ζ^{2}

into equation (25), it takes the following form

\frac{π \sqrt{ζ + ζ^{2}}}{2 \sqrt{2} ω} (1 - \frac{π^{2} ζ^{3}}{24 ω^{2}} - \frac{3 π^{4} ζ^{4}}{640 ω^{4}} \dots) = 1 .

(27)

Neglecting all higher-order terms of $ζ$ from equation (27), $ω$ is approximated as

ω = \frac{π \sqrt{ζ}}{2 \sqrt{2}} = \frac{π}{2 \sqrt{2 + a^{2}}} .

(28)

The proposed method

Consider that a polynomial-type series solution of equation (12) can be taken as

u (τ) = b + B_{2} τ^{2} + B_{4} τ^{4} + B_{6} τ^{6} + B_{8} τ^{8} + \dots,

(29)

where

B_{2}, B_{4}, B_{6}, B_{8},

etc. represent unknown coefficients that need to be determined.

Equation (12) simplifies to

ω^{2} u^{″} (\sqrt{1 + u^{2}}) + u = 0, u (0) = b and u^{″} = 0 .

(30)

By differentiating equation (29) with respect to

τ

and putting the values of

u^{″} (τ)

and

u (τ)

into equation (30), it becomes

ω^{2} (2 B_{2} + 12 B_{4} τ^{2} + {30 B}_{6} τ^{4} + 56 B_{8} τ^{6} + \dots) \times

(\sqrt{1 + {(b + B_{2} τ^{2} + B_{4} τ^{4} + B_{6} τ^{6} + B_{8} τ^{8} + \dots)}^{2}}) +

(b + B_{2} τ^{2} + B_{4} τ^{4} + B_{6} τ^{6} + B_{8} τ^{8} + \dots) = 0 .

(31)

Now, by equating the coefficients of the like power of

τ^{2}, τ^{4}, τ^{6}, τ^{8}, \dots

on both sides, the following algebraic equations are derived

b + 2 B_{2} ω^{2} (\sqrt{1 + b^{2}}) = 0,

(32)

B_{2} + 12 B_{4} ω^{2} {(1 + b^{2})}^{3 / 2} = 0,

(33)

- 3 b {B_{2}}^{2} + 2 (1 + b^{2}) B_{4} + 60 B_{6} ω^{2} {(1 + b^{2})}^{5 / 2} = 0,

(34)

(- 1 + 4 b^{2}) {B_{2}}^{3} - 6 b (1 + b^{2}) B_{2} B_{4} + 2 {(1 + b^{2})}^{2} (B_{6} + 56 {(1 + b^{2})}^{3 / 2} B_{8} ω^{2}) = 0,

(35)

The unknown coefficients are determined by solving equations (32) to (35) as follows

B_{2} = - \frac{b}{2 ω^{2} \sqrt{1 + b^{2}}}, B_{4} = \frac{b}{24 ω^{4} {(1 + b^{2})}^{2}}, B_{6} = \frac{- b + 9 b^{3}}{720 ω^{6} {(1 + b^{2})}^{7 / 2}}, B_{8} = \frac{b - 99 b^{3} + 180 b^{5}}{40320 ω^{8} {(1 + b^{2})}^{5}}, \dots .

Now, putting the values of unknown coefficients (up to

B_{8}

) into equation (29), it becomes

u (τ) = b + \frac{(b - 99 b^{3} + 180 b^{5}) τ^{8}}{40320 {(1 + b^{2})}^{5} ω^{8}} + \frac{(- b + 9 b^{3}) τ^{6}}{720 {(1 + b^{2})}^{7 / 2} ω^{6}} + \frac{b τ^{4}}{24 {(1 + b^{2})}^{2} ω^{4}} - \frac{b τ^{2}}{2 \sqrt{1 + b^{2}} ω^{2}} .

(36)

The value of $u (τ)$ is zero, when $τ = \frac{π}{2}$ . Putting $τ = \frac{π}{2}$ into equation (36), it becomes

b + \frac{(b - 99 b^{3} + 180 b^{5}) π^{8}}{10321920 {(1 + b^{2})}^{5} ω^{8}} + \frac{(- b + 9 b^{3}) π^{6}}{46080 {(1 + b^{2})}^{7 / 2} ω^{6}} + \frac{b π^{4}}{384 {(1 + b^{2})}^{2} ω^{4}} - \frac{b π^{2}}{8 \sqrt{1 + b^{2}} ω^{2}} = 0 .

(37)

Now, substituting

ω = \frac{π}{η}

(where

η

is a new variable) into equation (37), it yields

f (η) = b - \frac{b η^{2}}{8 \sqrt{1 + b^{2}}} + \frac{b η^{4}}{384 {(1 + b^{2})}^{2}} + \frac{(- b + 9 b^{3}) η^{6}}{46080 {(1 + b^{2})}^{7 / 2}} + \frac{(b - 99 b^{3} + 180 b^{5}) η^{8}}{10321920 {(1 + b^{2})}^{5}} = 0

(38)

The value of $η$ can be determined from equation (38) when $b$ is known. Therefore, the approximate value of $η$ can be efficiently obtained via the following iterative formula, with the initial guess $η_{0} = 0$ .

Now, the frequency of the relativistic equation can be obtained from $ω = \frac{π}{η}$ , using the value of $η$ from equation (39). Therefore, the solution of the transform equation (12) can be obtained from equation (36).

Finally, the solution of the original relativistic equation (6) is obtained from the following relation between $x (τ)$ and $u (τ)$ [see the details in Appendix A]

x (τ) = \sqrt{a^{2} + 2 - 2 \sqrt{1 + {u (τ)}^{2}}}, where a is the amplitude

(40)

Results and discussion

A more efficient differential transform method (DTM) has been applied to solve the relativistic equation. Recently, the DTM²⁹ method was presented to solve the same problem, but it is a laborious task (see Sub-section 2.2). Earlier, the modified Lindstedt–Poincare method (MLP)⁵ was used by Yeasmin²¹ to solve relativistic equation, equation (6). The determination of such a solution is also laborious. Moreover, it does not nicely agree with the numerical solution (see Figures 1 and 2). Belendez²⁰ used the harmonic balance method to determine the frequency of the oscillator equation (6) but he applied only a fundamental term to find the solution of the equation by integrating equation (10). It is noted that equation (10) cannot be integrated when $u (τ)$ contains more than the fundamental harmonic term. In general, two or three harmonic terms are considered to solve a nonlinear differential equation. The disadvantage of this method is that the algebraic equations for the unknown coefficients are nonlinear (see Appendix B).

Figure 1.

Comparison between the modified Lindstedt–Poincare (MLP) method²¹ and numerical solution at $a = 5$ for the relativistic equation.

Figure 2.

Comparison between the modified Lindstedt–Poincare method²¹ (MLP) method and numerical solution at $a = 10$ for the relativistic equation.

On the other hand, the proposed method has been developed using an easier approach, and the solution nicely agrees with the numerical solution. The algebraic equations for the unknown coefficients are linear and simple.

The approximate frequencies and the solutions of the relativistic equation (i.e., equation (6)) obtained by the proposed method have been compared with the existing methods and numerical method. The frequencies of the relativistic equation obtained by the proposed method, along with the Belendez,²⁰ Yeasmin,²¹ and Alam et al.²⁹ methods and exact results, are presented in Table 1 for different values of

a

. From Table 1, it is observed that all methods perform well for low amplitudes, and the proposed method provides significantly better results than existing methods for oscillation with a large amplitude.

Table 1.

Comparison of the proposed approximate frequencies with exact and other existing frequencies of the relativistic equation.

Amplitude $a$	Frequencies $(ω)$ (error (%))
Amplitude $a$	Exact	Belendez²⁰	HBM	MLP²¹	Alam et al.²⁹	Proposed method
1	0.851301	0.847597 (0.435099)	0.851037 (0.031047)	0.851301 (0.000000)	0.906900 (6.531005)	0.849955 (0.158181)
5	0.301640	0.292327 (3.087455)	0.297969 (1.217014)	0.301935 (0.218804)	0.302300 (0.218760)	0.302020 (0.125905)
10	0.155498	0.150465 (3.236698)	$0 .$ 153413 (3.236698)	0.155789 (0.187141)	0.155532 (0.021684)	0.155522 (0.015184)
50	0.031403	0.030381 (3.254466)	0.0309770 (1.356558)	0.031479 (0.242015)	0.031403 (0.000062)	0.031403 (0.000051)
100	0.015706	0.015195 (3.253533)	0.015493 (1.356170)	0.015744 (0.241946)	0.015706 (0.000005)	0.015706 (0.000004)

where, Errors (%) represent the absolute relative percentage error.

The solutions of equation (6) calculated via the proposed method together with different existing methods and the numerical method (Runge–Kutta 4^th order) are shown in Figures 1-6, for different values of a. Figures 1 and 2 illustrate comparison of the relativistic equation between the numerical solution and Yeasmin²¹ method for $a = 5$ and $a = 10$ , respectively. It is observed that the solution of the modified Lindstedt–Poincare method⁵ exhibits a triangular waveform, indicating non-smooth behavior.

Figure 3.

Comparison of the Belendez²⁰ method and harmonic balance method (HBM) solution with the numerical solution at $a = 5$ for the relativistic equation.

Figure 4.

Comparison of the Belendez²⁰ method and harmonic balance method (HBM) solution with the numerical solution at $a = 10$ for the relativistic equation.

Figure 5.

Comparison of the proposed method and Alam et al.²⁹ method solution with the numerical solution at $a = 5$ for the relativistic equation.

Figure 6.

Comparison of the proposed method and Alam et al.²⁹ method solution with the numerical solution at $a = 10$ for the relativistic equation.

Figures 3 and 4 show a comparison of the numerical solution of the relativistic equation with that obtained by using the Belendez²⁰ method and the harmonic balance method, respectively, for $a = 5$ and $a = 10$ . It is observed that both the Belendez²⁰ and harmonic balance method solutions deviate from the numerical solution.

Figures 5 and 6 demonstrate a comparison of the solutions among the proposed method (using term up to $τ^{8}$ ), Alam et al.²⁹ differential transform method (using term up to $φ^{10}$ ), and the numerical solution, respectively, for $a = 5$ and $a = 10$ . From the Figures 5 and 6, it is seen that, both methods provide the same results as the numerical results, but the proposed method takes a lower number of terms in the solution than Alam et al.²⁹ method.

Comparing the determination of various solution and seeing all the figures and the results of the Table 1, it may be decided that the proposed method is efficient than all other methods described in this article.

Footnotes

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 iDs

Mostafa Amir Faishal

Mohammad Kamrul Hasan

Appendix

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A more efficient differential transform method for solving relativistic equation

Abstract

Keywords

Introduction

Derivation of the relativistic equation

Transformation of the non-dimensional relativistic equation 20

Methodology

The classical differential transform method (DTM) 22

Alam et al. 29 Differential transform method

The proposed method

Results and discussion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Appendix

References

Transformation of the non-dimensional relativistic equation²⁰

The classical differential transform method (DTM)²²

Alam et al.²⁹ Differential transform method