Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable order

Devendra Chouhan,Vinod Mishra,H.M Srivastava

doi:10.1016/j.rinam.2021.100146

Abstract

In this paper, generalized fractional-order Bernoulli wavelet functions based on the Bernoulli wavelets are constructed to obtain the numerical solution of problems of anomalous infiltration and diffusion modeling by a class of nonlinear fractional differential equations with variable order. The idea is to use Bernoulli wavelet functions and operational matrices of integration. Firstly, the generalized fractional-order Bernoulli wavelets are constructed. Secondly, operational matrices of integration are derived and utilize to convert the fractional differential equations (FDE) into a system of algebraic equations. Finally, some numerical examples are presented to demonstrate the validity, applicability and accuracy of the proposed Bernoulli wavelet method.

Highlights

Fractional calculus is an old mathematical topic from 17th century [1]
Fractional calculus are increasingly used in modeling of practical problems in various areas of engineering and physics such as continuum and statistical mechanics [2], fluid mechanics [3] dynamic of viscoelastic materials [4], econometrics [5], electromagnetism [6], propagation of spherical flames [7], Ψ -Hilfer problem [8], fractional PDE-constrained optimization problems [9] and in many other fields [10,11,12,13,14,15,16,17,18]
Extensive research has been performed on the development of numerical methods for the solution of fractional differential equations (FDE) such as variational iteration method [20], Adomian decomposition method [21], fractional differential transform method [22], operational approach [23,24,25] and wavelet methods like Chebyshev wavelet method [26], Legendre wavelet method [27], Haar wavelet method [28,29,30], Shannon wavelet method [31], Taylor wavelet collocation method [32] and cubic B-spline wavelet collocation method [33]

Summary

Introduction

Fractional calculus is an old mathematical topic from 17th century [1]. Fractional calculus are increasingly used in modeling of practical problems in various areas of engineering and physics such as continuum and statistical mechanics [2], fluid mechanics [3] dynamic of viscoelastic materials [4], econometrics [5], electromagnetism [6], propagation of spherical flames [7], Ψ -Hilfer problem [8], fractional PDE-constrained optimization problems [9] and in many other fields [10,11,12,13,14,15,16,17,18]. Extensive research has been performed on the development of numerical methods for the solution of FDEs such as variational iteration method [20], Adomian decomposition method [21], fractional differential transform method [22], operational approach [23,24,25] and wavelet methods like Chebyshev wavelet method [26], Legendre wavelet method [27], Haar wavelet method [28,29,30], Shannon wavelet method [31], Taylor wavelet collocation method [32] and cubic B-spline wavelet collocation method [33]. Wavelet theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering and other science disciplines. It has many applications in signal analysis, image processing, data compression, detection of submarine and aircrafts, prediction of earthquake and early detection of breast cancer etc

Objectives

Results

Conclusion