The construction of operational matrices of integral and fractional integral using the flatlet oblique multiwavelets

Abstract

The main aim of this paper is to introduce the operational matrices of integral and fractional integral using the flatlet oblique multiwavelets. The operational matrices of integral and fractional integrals for flatlet scaling functions and wavelets are presented and are utilized to reduce the solution of the Abel integral equations of the first and the second kinds to the solution of algebraic equations. The main characteristic behind our approach in using this technique is that only a small number of flatlet oblique multiwavelets is needed to obtain a satisfactory result. Illustrative examples reveal that the present method is very effective and convenient.

Application of flatlet oblique multiwavelets to solve the fractional stochastic integro‐differential equation using Galerkin method

“…The $$$ m+1 $$$ scaling functions and the $$$ m+1 $$$ wavelets of the FMWS [25, 28] with multiplicity $$$ m+1 $$$ are defined on $$$ \left[0,1\right] $$$. When $$$ m=0 $$$, the simplest case of the flatlet family in the form of the Haar wavelets appears.…”

“…The integral of $$$ \tilde{\boldsymbol{\Theta}}(x) $$$ in () can be written as $$$$ {\int}_0^1\tilde{\boldsymbol{\Theta}}(x) dx=\mathbf{I}\tilde{\boldsymbol{\Theta}}(x), $$$$ where $$$ \mathbf{I} $$$ recalls $$$ N\times N $$$ operational matrix of integral for the BFMS on $$$ \left[0,1\right] $$$ and the $$$ ij $$$th entry of $$$ \mathbf{I} $$$ is resulted as follows (for more details, see [25]): $$$$ {\left[\mathbf{I}\right]}_{i,j}=\left\langle {\boldsymbol{\Theta}}_i(x),{\int}_0^1{\tilde{\boldsymbol{\Theta}}}_j(x) dx\right\rangle, i,j=1,\dots, N. $$$$ …”

A New Numerical Algorithm for Solving a Class of Fractional Advection-Dispersion Equation with Variable Coefficients Using Jacobi Polynomials

“…In this section, we give some definitions and properties of the fractional calculus (see, e.g., [9,18,33,34]) and Jacobi polynomials (see, e.g., [35][36][37]).…”

High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation

“…The fractional advection-dispersion equation provides a useful description of chemical and contaminant transport in heterogeneous aquifers [2,4], abnormal mass absorption in solids [22] and densities of plumes in spread proportionally to time-dependent of fractional order [3]. Although there is a plenty of research in one dimensional Riesz space fractional advection-dispersion equation (for example, see [1,8,9,13,14,15,17,21,23,24,28,30]), there are few works in the two dimensional case [6,29,31]. In this research, we numerically solve the general two dimensional Riesz space fractional advection-dispersion equation using modified Crank-Nicolson Alternating Direction Implicit (ADI) method.…”