Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets

Abstract

Abstract:Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.

Solving Poisson’s Equations with fractional order using Haarwavelet

“…The simplicity in building the wavelet bases from any function which use only two operations translation and dilation [3], this can be easily seen in Haar wavelet.The simple form of the mother function in Haar wavelet as we see below makes the processes of dilation and translation an easy work and the introduced wavelet family is orthogonal not only linearly independent. Although, the wavelet function appeared in 1910, their use in the solution of differential equations does not appear until recently [4][5][6], last twenty years.In 2017 Kaoud and El Dewaik, [7] have used Haar wavelet technique to solve Poisson's equation on a unit square domain with collocation points j/16 , j = 1, 3, . .…”