Reproducing Kernel method for the solution of nonlinear hyperbolic telegraph equation with an integral condition

Abstract

In this article, an iterative method is proposed for solving nonlinear hyperbolic telegraph equation with an integral condition. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation u n (x, t) of the exact solution u(x, t) is obtained and is proved to converge to the exact solution. Moreover, the partial derivatives of u n (x, t) are also convergent to the partial derivatives of u(x, t). Some numerical examples have been studied to demonstrate the accuracy of the present method. Results obtained by the method have been compared with the exact solution of each example and are found to be in good agreement with each other.

A New Semi-Analytical Solution of the Telegraph Equation with Integral Condition

“…Thus from (13), we can obtain an analytical approximate solution for the problem. To assess the impact ofh on the convergence of the obtained approximate solution, we Table 1; the last row in this table shows the obtained result by the author of [16]. Clearly, we can observe that the approximate solutions obtained whenh = −1 are more accurate than the approximate solutions obtained form another choice ofh.…”

“…As we observe, there is a very good agreement between the approximate solution obtained by HAM and the exact solution. 2.9641e-3 2.4364e-4 4.1228e-5 6.6278e-6 4.2167e-7 v − v 225 ∞ presented in [16] 8e-4 Table 1. Numerical results for Example 1.…”

“…In [15], the authors investigated this problem and discussed the existence and uniqueness of the solution of this important problem by using the Rothe method. In [16], the author proposed an iterative method for solving this problem. Recently, Salkuyeh and Roohani in [17] applied the variational iteration method for solving an special case of this problem (the problem with linear forcing).…”

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

“…To achieve this goal, we must first construct reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions, and then implement RKM without Gram–Schmidt orthogonalization process on the problem (1). Some numerical methods such as RKM with Gram–Schmidt orthogonalization process (see [19–22]) have already been used by researchers for parabolic partial differential equation (PPDE) with nonlocal conditions [23–26] and indeed here we improve the approximate solutions to the problem (1) with more accuracy. Additionally, we prove the stability theorem of the method and provide error analysis for this technique.…”

Numerical Solutions of the Second-Order One-Dimensional Telegraph Equation Based on Reproducing Kernel Hilbert Space Method

“…Several test problems were given, and the results of numerical experiments were compared with analytical solutions to confirm the good accuracy of their scheme. Yao [8] investigated a nonlinear hyperbolic telegraph equation with an integral condition by reproducing kernel space at = = 0. Yousefi presented a numerical method for solving the one-dimensional hyperbolic telegraph equation by using Legendre multiwavelet Galerkin method [9].…”