A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems

Abstract

In this work, the authors developed two new B-spline collocation algorithms based on cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic partial differential equations (PDEs) with Dirichlet and Neumann boundary conditions. In the first algorithm, cubic trigonometric B-spline functions are directly used for approximate solutions of parabolic PDEs with Neumann boundary conditions. But, the Dirichlet boundary conditions cannot be handled directly by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and the second algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithms reduced the parabolic problem into a system of first-order nonlinear ordinary differential equations (ODEs) in time variable. Then, strong stability-preserving-Runge–Kutta3 (SSP-RK3) scheme is used to solve the obtained system. Some well-known parabolic problems are solved to check the accuracy and efficiency of the proposed algorithms. The algorithms can be extended to solve multidimensional problems arising as model equations in physical, chemical and biophysical phenomenon.