Solutions to nonlinear pseudo hyperbolic partial differential equations with nonlocal conditions by

Abstract

Keywords: Nonlinear Pseudo Hyperbolic Equations; Partial Differential Equation; Residual Power Series Method; Analytical Solutions; Exact Solutions

Introduction

Firstly, we consider the generalised nonlinear pseudo-hyperbolic equation with nonlocal conditions as follows: 𝑢𝑡𝑡 (𝑡,𝑥) − 𝑢(𝑡,𝑥)𝑢𝑡𝑥𝑥(𝑡,𝑥) − 𝑢𝑥𝑥(𝑡,𝑥) − 𝑓(𝑡,𝑥) = 0, 𝑥 ∈ (0,𝑋), 𝑡 ∈ (0,𝑇).

Linear and nonlinear partial differential equations describe a range of phenomena in different fields of sciences such as engineering and physics.

*Corresponding author. *E-mail address: sadiq.taha@uod.ac This paper was recommended for publication in revised form by Regional Editor Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Sigma J Eng Nat Sci, Vol. 41, No. 3, pp. 488−492, June, 2023

The exact solutions of this type of problems are rare in the literature, because it is not easy to find accurate solutions of nonlinear problems. Therefore, the analytical and numerical methods are used to find the approximation solution. Recently, many authors have been interested in testing linear and nonlinear partial and fractional differential equations using various numerical and analytical methods, among them the residual power series method [1], Adomian decomposition method [2], the finite difference method [3], the Variational iteration method [4], the homotopy analysis method [5], the reproducing kernel Hilbert space method [6] and the multiple Exp-function method [7], has been applied to solve linear and nonlinear partial differential equations. The residual power series (RPS) method was first proposed by Arqoub [8] and it is an accurate and convenient technique for constructing power series solutions of differential equations and RPSM has been shown to be a suitable and effective method in its applications, and has been widely used to solve different types of problems [9-15]. The Hyperbolic partial differential equations model the longitudinal vibrations of structures, such as beams, buildings and machines [2,16-18] and have been studied to model real engineering problems [19-21]. The pseudo-hyperbolic equations of the form (1) is a type of high order partial differential equation with mixed partial derivatives concerning space and time, which describes many physical phenomena such as, longitudinal vibrations, heat and mass transfer, nerve conduction and reaction-diffusion [18,23,25]. In recent years, some authors have been applied the numerical and analytical methods for solving the pseudohyperbolic equation. The authors in [22] obtained the approximate analytical solutions of the pseudohyperbolic equation via residual power series method. The numerical approximation scheme based on the H1-Galerkin mixed finite element method to pseu-dohyperbolic equation was constructed in [23]. The authors in [18] studied pseudo-hyperbolic equations and provided a comprehensive understanding of hyperbolic and pseudo-hyperbolic operators arising in the theory of longitudinal and lateral vibrations of elastic bars. In [24] the authors provided the sufficient condition for the nonexis-tence of weak solutions to the nonlinear pseudo-hyperbolic equation in the Heisenberg theory. In [26] the authors was discussed the splitting positive definite mixed finite ele-ment methods for pseudohyperbolic equations and the two least-squares Galerkin finite element schemes was for-mulated to solve pseudo hyperbolic equations in [27]. The aim of this article is to investigate the analytical solution of nonlinear pseudohyperbolic partial differential equation that depends on the nonlocal condition by an analytical method namely residual power series (RPS) method. The rest of this paper organised as follows, in the sec-ond section the analysis of the (RPS) method is given. In the third section applications of (RPS) method to nonlinear

Pseudo-Hyperbolic Equation are presented. Finally, in the last section conclusion is presented. Analysis of RPSM In this section, we give the basic information about the proposed method for solving nonlinear Pseudo-Hyperbolic Equation, the solutions of equations (1) and (2) in (RPS) method are expressed as the power series expansion at the initial point t = 0. First, we suppose that the solutions take expansion (3) now, we determine uk(x,t) to indicate the kth truncated series of the u(t,x) as follows (4) It is clear that u(t,x) verify the initial condition (2), therefore the approximate solutions to zeroth via (RPS) method for u(t,x) is (5) However, the initial condition expressed by approximate solution from equation (4), then the first approximate solutions of (RPS) method must be (6) Consequently, the expanding power series of Equation (4) can be expressed as (7) To obtain the values of coefficients fm(x), for m = 2,3,4, ... k by the residual power series method, in the series expanding of equation (7), the residual functions could be defined as (8) Consequently, the kth residual functions Resu(x,t), are of the following form (9) from the equation (9), we write , where and t ≥ 0 Thus, 0 and s = 0, ..., k as shown in [12].

Sigma J Eng Nat Sci, Vol. 41, No. 3, pp. 488−492, June, 2023

For obtaining the coefficients fm(x), where m = 2, 3, 4, ... k, we use the following procedures; substituting kth trun-

cated series of u(t, x) in equation (9), and applying the formula

0, then the following equation is equalised it to zero. Finally to obtain the form of other coefficients, we need to solve the following equation , for more details see [22]:

Using the iterative formula of (RPS) method for m = 0, 1, 2, ... we get (13)

(10) In this way, we can get all the required coefficient of equations (1) and (2) for a series of multiple exponents. Application of the (RPS) Method to Pseudo-Hyperbolıc Equations This section presents the applications of (RPS) method for solving nonlinear pseudo-hyperbolic equations depends on nonlocal conditions. for this, We test some examples to show the accuracy and efficacy of the proposed method. Example 1. Consider the nonlinear pseudo-hyperbolic equation (11)

By using (6) and from the initial condition (12), we get u1(t, x) = sinx, substitute u1(t, x) in formula(13), we obtain u2(t, x) = 0, in the same way, we can obtain u3(t, x), u4(t, x), ..., and substitute in formula (7), then we get the exact solution: (14) Example 2. Consider the nonlinear pseudo-hyperbolic equation

Figure 1. Gives the exact solution of u(t,x) for example 1., at 0 ≤ t ≤ 1 and 0 ≤ x ≤ π.

Sigma J Eng Nat Sci, Vol. 41, No. 3, pp. 488−492, June, 2023

Conclusion

Using the iterative formula of (RPS) method for m = 0, 1, 2, ... we get

In this paper, a new results for nonlinear pseudo-hyperbolic equations depend non-local conditions by using (RPS) method are given, this method based on the Taylor series formula with residual error function. The method has excellent and accurate results for this equations. Simulations are shown for the obtained results. Therefore, the RPSM method is notables because it gives the exact solution to nonlinear pseudo-hyperbolic equations. The obtained result reveal that the accuracy and fast convergence for the proposed method.

Acknowledgements

The authors have no direct funding and support for this work.

Data Availability Statement

by the same way for u3(t, x), u4(t, x), ..., and using the Taylor series formula, then it gives exact solution (18)

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Figure 2. Illustrates the exact solution of u(t,x) for example 2., at 0 ≤ t ≤ 1 and 0 ≤ x ≤ π.

Sigma J Eng Nat Sci, Vol. 41, No. 3, pp. 488−492, June, 2023

Conflict Of Interest

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

Ethics

There are no ethical issues with the publication of this manuscript.

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