Abstract
The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.
Highlights
IntroductionThe problem of solving parabolic partial differential equation in its very simple case
The problem of solving parabolic partial differential equation in its very simple case has been considered by many authors, it is the mathematical model of many physical, engineering, and finance problems
It is well known that the solution of this simple problem requires one initial condition and two boundary conditions
Summary
The problem of solving parabolic partial differential equation in its very simple case. The method of solution of this simple problem depends on the nature of the boundary conditions as well as the initial condition. Equation (2) has many advantages in modeling real problems than equation (1) appearance of fractional time derivatives, non-homogenous terms (in the form of integrals). It is generally accepted that differential equations consider the local effects in modeling physical phenomenon, integral equations consider the global behaviors [6], while the fractional derivatives consider the memory and hereditary behaviors [2, 7, 8]. The technique is to approximate the highest derivative of the differential equation with finite Haar wavelet series. Integrate this approximation to get the lower order derivatives in the equation. Many authors use this technique to solve the differential or integral equations [14, 15, 16, 17, 18]
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