An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

and Applied Analysis 3 and the boundary conditions, u (A, t) = k 1 (t) , u (B, t) = k 2 (t) , V (A, t) = k 3 (t) , V (B, t) = k 4 (t) , t ∈ [0, T] . (10) Starting with the transformations x = ((2/(B − A))y) + ((A + B)/(A − B)), w(x, t) = u(y, t), and z(x, t) = V(y, t). Problem (8)–(10) will be a new problem in the spatial variable x ∈ [−1, 1]. This transformation enable us to use the Jacobi collocation method on [−1, 1], D 2 t w (x, t) = γw (x, t) z (x, t) ( 2 (D y w (x, t) + D y z (x, t)) B − A +D t w (x, t) + D t z (x, t) ) + 4g 1 (y, t)D y w (x, t) (B − A) 2 + g 2 (y, t) , D 2 t z (x, t) = δw (x, t) z (x, t) ( 2 (D y w (x, t) + D y z (x, t)) B − A +D t w (x, t) + D t z (x, t) ) + 4g 3 (y, t)D y z (x, t) (B − A) 2 + g 4 (y, t) , (y, t) ∈ [A, B] × [0, T] , (11) subject to a new set of initial and boundary conditions, w (x, 0) = f 5 (x) , D t w (x, 0) = f 7 (x) , z (x, 0) = f 6 (x) , D t z (x, 0) = f 4 (x) , x ∈ [−1, 1] , (12) w (−1, t) = k 1 (t) , w (1, t) = k 2 (t) , z (−1, t) = k 3 (t) , z (1, t) = k 4 (t) , t ∈ [0, T] . (13) Now, we are interested in using the J-GL-C method to transform the previous coupled PDEs into system of ODEs. In order to do this, we approximate the spatial variable using J-GL-C method at some nodal points. The node points are the set of points in a specified domain where the dependent variable values are approximated. In general, the choice of the location of the node points is optional, but taking the roots of the Jacobi orthogonal polynomials referred to as Jacobi collocation points gives particularly accurate solutions for the spectral methods. Now, we outline the main step of the JGL-Cmethod for solving couples hyperbolic problem. Let us expand the dependent variable in a Jacobi series,

Exact solutions for initial value problem for some nonconservative hyperbolic systems are presented in [29], while the analytical study of variable coefficient mixed hyperbolic partial differential problems is discussed in [30].The solitary and periodic wave solutions have been studied for some kinds of hyperbolic Klein-Gordon equations in [31,32].
Other numerical methods based on the boundary integral equation [33] and numerical integration techniques [34] are used to numerically solve different types of hyperbolic partial differential problems.In [35,36], finite difference scheme is considered to numerically solve hyperbolic equations.Pseudospectral methods are used in [37][38][39][40] to solve Klein-Gordon equations.In [41], Dehghan and Shokri used the radial basis functions to solve a two-dimensional Sine-Gordon equation; moreover in [42] they developed numerical scheme to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity using collocation points and approximating the solution using Thin Plate Splines and RBFs.
There are no results on Jacobi-Gauss-Lobatto collocation (J-GL-C) method for solving nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary and nonlocal conditions.Therefore, the objective of this work is to present this method to numerically solve four nonlinear coupled hyperbolic PDEs with variable coefficients.By using collocation method, exponential convergence for the spatial variables can be achieved to approximate the solution of PDE.
The computerized mathematical algorithm is the main key to apply this method for solving the problem.Moreover, the nonlocal conservation conditions are efficiently treated by Jacobi-Gauss-Lobatto quadrature rule at ( + 1) nodes to obtain a system of ODEs in time and then proper initial value software can be applied to solve this system of ODEs.Several illustrative problems with various kinds of exact solutions such as triangular, soliton, and exponentialtriangular solutions are presented for demonstrating the high accuracy of this scheme.Moreover, with the freedom of selecting the Jacobi indexes  and , the scheme can be calibrated for a wide variety of problems.Finally, the accuracy of the proposed method is demonstrated by solving some test nonlinear problems.
A brief outline of this paper is as follows.We present some properties of Jacobi polynomials in the next section.The third section is divided into two subsections: the first one deals with coupled nonlinear hyperbolic PDE with initial-boundary conditions.The numerical treatment of solve initial-nonlocal conservation conditions is developed in Section 3.2.In Section 4 the proposed method is applied to four different test problems to show the accuracy of our method.In the last section, we present some observations and conclusions.

The Problem and the Numerical Algorithm
In this section, we approximate the solution of coupled nonlinear hyperbolic types equations with two different kinds of boundary conditions for space variable by using the Jacobi collocation method.
subject to a new set of initial and boundary conditions, Now, we are interested in using the J-GL-C method to transform the previous coupled PDEs into system of ODEs.In order to do this, we approximate the spatial variable using J-GL-C method at some nodal points.The node points are the set of points in a specified domain where the dependent variable values are approximated.In general, the choice of the location of the node points is optional, but taking the roots of the Jacobi orthogonal polynomials referred to as Jacobi collocation points gives particularly accurate solutions for the spectral methods.Now, we outline the main step of the J-GL-C method for solving couples hyperbolic problem.Let us expand the dependent variable in a Jacobi series, And, in virtue of ( 6)-( 7), we evaluate   () and   () by The Jacobi-Gauss-Lobatto quadrature has been used to evaluate the previous integrals accurately.For any we have that For any positive integer ,   [−1, 1] stands for the set of polynomials of degree at most ,  (,) , (0 ≤  ≤ ) and  (,) , (0 ≤  ≤ ) are used as the nodes and the corresponding Christoffel numbers in the interval [−1, 1], respectively.Thanks to (6), the coefficients   () in terms of the solution at the collocation points can be approximated by Due to (17), the approximate solution can be written as Furthermore, if we differentiate (18) once and evaluate it at the first  + 1 Jacobi-Gauss-Lobatto collocation points, it is easy to compute the first spatial partial derivative of the numerical solution in terms of the values at these collocation points as where Accordingly, one can obtain the second spatial partial derivative as where In the proposed J-GL-C method the residual of ( 11) is set to zero at  − 1 of Jacobi-Gauss-Lobatto points; moreover, the boundary conditions (13) will be enforced at the two collocation points −1 and 1.Therefore, the approximation of ( 11)-( 13) is where This approach provides a (2 − 2) system of second order ODEs in the expansion coefficients   (),   (), with the following initial conditions: or in matrix notation as where The system of second order ( 27)-( 28) can be solved by using diagonally implicit Runge-Kutta-Nyström (DIRKN).

Initial-Nonlocal Conservation Conditions.
Here, we will implement the J-GL-C algorithm for the coupled nonlinear hyperbolic type equations with nonlocal conditions: subject to the initial conditions, and the boundary conditions, The problem now is how to deal with the nonlocal conditions (37).For this purpose, let us introduce a collocation treatment for the integral conservation conditions (37) as The above equations may be rearranged as or briefly where Consequently,   () and   () are expressed as the following expansion of   () and   (),  = 1, . . ., : Based on the information included in this subsection and the recent one, we obtain the following system of ODEs: with the following initial conditions: (0) =  5 ( (,) ,n ) , ẇ  (0) =  7 ( (,) , ) ,   (0) =  6 ( (,) , ) , ż  (0) =  8 ( (,) , ) , where  0 ,   ,  0 , and   are given in (36) and (42).

Test Problems
We test the numerical accuracy of the proposed method by introducing four test problems with different types of exact solutions.

Triangular Solution.
As a first example, we consider the coupled nonlinear hyperbolic equation ( 8) with the following functions:  ( The exact solutions of this problem are The absolute errors in the given tables are  (, ) =      (, ) − ũ (, )     , where (, ) and ũ(, ) are the exact and approximate solutions at the point (, ), respectively.Moreover, the maximum absolute error is given by The root mean square (RMS) and   errors may be given by Maximum absolute, root mean square, and   errors of (45) are introduced in Table 1 using J-GL-C method with three different choices of , , and  in the interval [0, 1].The approximate solutions ũ and Ṽ of problem (45) have been plotted in Figures 1 and 2, with values of parameters listed in their captions.Moreover, we plot the curves of approximate and exact solutions of ũ at different values of  and  in Figures 3 and 4. Again, the curves of approximate and exact solutions of Ṽ at different values of  and  are displayed in Figures 5  and 6.

Soliton Solution.
Secondly, consider the coupled nonlinear hyperbolic equation ( 8) with the following functions: subject to The exact solutions are Table 2 shows the accurate results for maximum absolute, root mean square, and   errors of (51) for various choices of , , and  in the interval [0, 1].Figures 7 and 8 show that the absolute errors  1 and  2 are very small with values of parameters listed in their captions.We also plot the curves of approximate and exact solutions of ũ at different values of  and  in Figures 9 and 10.Moreover, in Figures 11 and 12, the approximate and exact solutions of Ṽ are plotted at different values of  and .
Maximum absolute, root mean square, and   errors of (57) are introduced in  solutions for problems with nonlocal conservation conditions are in good agreement with the exact solutions.

Table 3 ,
for different choices of Jacobi parameters; even we use limited values of .The approximate solutions ũ and Ṽ of problem (55) are plotted in Figures13 and 14with values of parameters listed in their captions.In addition, Figures15 and 16present the approximate and exact solutions of ũ(, ); moreover, the corresponding figures for Ṽ(, ) at parameters listed in their captions are displayed in Figures17 and 18

Table 4
using J-GL-C method with various choices of , , and  in the interval [0, 1].From numerical results of this table, it can be concluded that the numerical