This paper studies the dynamics of nonlinear singular heat conduction model of the human head, and presents a new fractional heat conduction model for the human head taking into account the effect of febrifuge. The Taylor series and two-scale transform method are developed for solving the fractional model. The correctness and effectiveness of the proposed method are also ascertained through one example. The results show that the proposed method is simple but effective.

A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge

In this paper, a potentially useful new method based on the Gegenbauer wavelet expansion, together with operational matrices of fractional integral and block-pulse functions, is proposed in order to solve the Bagley–Torvik equation. The Gegenbauer wavelets are generated here by dilation and translation of the classical orthogonal Gegenbauer polynomials. The properties of the Gegenbauer wavelets and the Gegenbauer polynomials are first presented. These functions and their associated properties are then employed to derive the Gegenbauer wavelet operational matrices of fractional integrals. The operational matrices of fractional integrals are utilized to reduce the problem to a set of algebraic equations with unknown coefficients. Illustrative examples are provided to demonstrate the validity and applicability of the method presented here.

An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation

We provide a numerical method to solve a certain class of fractional differential equations involving $$\psi $$
 -Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce $$\psi $$
 -shifted Legendre polynomial basis, and we derive an explicit formula for the $$\psi $$
 -fractional integral of $$\psi $$
 -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.

/pdf/a-numerical-study-of-fractional-relaxation-oscillation-gijrgug33c.pdf

A numerical study of fractional relaxation–oscillation equations involving $$\psi $$ ψ -Caputo fractional derivative

In the present paper, we numerically simulate fractional-order model of the Bloch equation by using the Jacobi polynomials. It arises in chemistry, physics and nuclear magnetic resonance (NMR). It also arises in magnetic resonance imaging (MRI) and electron spin resonance (ESR). It is used for purity determination, provided that the molecular weight and structure of the compound is known. It can also be used for structural determination. By the study of NMR, chemists can determine the structure of many compounds. The obtained numerical results are compared and simulated with the known solutions. Accuracy of the proposed method is shown by providing tables for absolute errors and root mean square errors. Different orders of the time-fractional derivatives results are illustrated by using figures.

https://www.mdpi.com/2076-3417/10/8/2850/pdf

Numerical Simulation for Fractional-Order Bloch Equation Arising in Nuclear Magnetic Resonance by Using the Jacobi Polynomials

A study of the fractional-order mathematical model of diabetes and its resulting complications

In this paper, a new operational matrix method based on Haar wavelets is proposed to solve linear and non-linear differential equations of fractional order. Contrary to wavelet operational methods available in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The main characteristics of our approach is that it converts fractional differential equations to system of algebraic equations and does not require the inverse of the Haar matrices. Illustrative examples are included to demonstrate the validity and applicability of the present method. Moreover, special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.

Numerical Solution of Fractional Differential Equations Using Haar Wavelet Operational Matrix Method

2 Abstract: In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The proper- ties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive.

APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE Numerical solution of singularly perturbed problems using Haar wavelet collocation method

In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive.

Numerical solution of singularly perturbed problems using Haar wavelet collocation method

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics Contrar

R. Abass

Papers

An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation

Numerical Solution of Fractional Differential Equations Using Haar Wavelet Operational Matrix Method

APPLIED & INTERDISCIPLINARY MATHEMATICS | RESEARCH ARTICLE Numerical solution of singularly perturbed problems using Haar wavelet collocation method

Numerical solution of singularly perturbed problems using Haar wavelet collocation method

Generalized wavelet collocation method for solving fractional relaxation-oscillation equation arising in fluid mechanics