Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation

Abstract

The generalized nonlinear Klien-Gordon equation is important in quantum mechanics and related fields. In this paper, a semi-implicit approach based on hybrid cubic B-spline is presented for the approximate solution of the nonlinear Klien-Gordon equation. The usual finite difference approach is used to discretize the time derivative while hybrid cubic B-spline is applied as an interpolating function in the space dimension. The results of applications to several test problems indicate good agreement with known solutions.

Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations

“…The approximate solution ( , ) to the exact solution exc ( , ) can be expressed as follows [59][60][61][62][63][64]:…”

“…Further two equations are included in the resulting system to obtain a unique solution of the problem using the boundary conditions given in (21). The initial vector 0 can be obtained from initial condition given in (2) [59][60][61][62][63][64]. Thus, the resulting system becomes a matrix system of dimension ( +3)×( +3) which is a tridiagonal system that can be solved by Thomas algorithm [65][66][67].…”

“…and applying same procedure for nonlinear term as in [64], the GBH equation can be converted to the following:…”

Hybrid Cubic B-Spline Method for Solving Non-Linear Two-Point Boundary Value Problems

“…Hybrid cubic B-spline basis function is established by using a linear combination of the cubic B-spline basis function and trigonometric cubic B-spline basis [20].…”

“…The main purpose of this study is to apply a hybrid cubic B-spline function in solving equation (1). The hybrid cubic B-spline method used is basically that of [20] for solving a generalized nonlinear Klien-Gordon equation for which the results are more accurate compared with some other methods. Therefore we seek to apply HCBSM for solving equation (1).…”