Numerical approximation of time-fractional nonlinear partial integro-differential equation using fractional Euler and cubic trigonometric B-Spline methods

Abstract

Nonlinear mathematical problems arise due to existence of important complex nonlinear phenomena in engineering and science. In this article, a class of time-fractional nonlinear parabolic partial integro-differential equations is solved numerically by combination of fractional Euler and cubic trigonometric B-spline collocation methods. Backward finite difference formula is employed for time-fractional Caputo derivative to get an unconditional stable scheme. The memory(integral) term is evaluated using a second order quadrature rule. Fractional Euler method for Caputo derivative is used in computing the nonlinear memory term. At each time level, cubic trigonometric B-spline functions are applied to obtain the solution in spatial dimension which reduces the problem to a system of algebraic equations. This method has the ability to handle any kind of nonlinearity without using iterative processes. Efficiency and reliability of the current method is analyzed for the fractional-order via three highly nonlinear test problems with variable coefficients. The rate of convergence of the proposed method is also computed in temporal and spatial dimensions.