An accurate collocation method for distributed order time fractional nonlinear diffusion wave equation with error analysis

Abstract

The distributed-order fractional nonlinear diffusion-wave problem is a mathematical model that combines the concepts of fractional calculus and nonlinear diffusion-wave equations. It involves the use of distributed-order fractional operators, which generalize the traditional constant-order fractional operators by allowing the order of the derivative to vary over a range of values. This method works especially well for modeling complex systems whose behavior is affected by memory and nonlocal effects that happen across several scales. The objective of this article is to offer an appropriate numerical method for treating this problem. In order to achieve this, we dealt with the integral terms in the main equation using the Newton–Cotes quadrature rule. The problem reduces to a nonlinear system of equations through the computation of operational matrices. With the Levenberg–Marquardt algorithm as an option, the resulting system had been solved using Matlab’s fsolve tool. The analysis of the scheme and the function approximation have been thoroughly covered. Some test problem provided to compare the method with existing one. Additionally, the effect of collocation points on the numerical solution’s accuracy has been investigated.