Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Abstract
Purpose
The purpose of this paper is to obtain a numerical scheme for finding numerical solutions of linear and nonlinear Hadamard-type fractional differential equations.
Design/methodology/approach
The aim of this paper is to develop a numerical scheme for numerical solutions of Hadamard-type fractional differential equations. The classical Haar wavelets are modified to align them with Hadamard-type operators. Operational matrices are derived and used to convert differential equations to systems of algebraic equations.
Findings
The upper bound for error is estimated. With the help of quasilinearization, nonlinear problems are converted to sequences of linear problems and operational matrices for modified Haar wavelets are used to get their numerical solution. Several numerical examples are presented to demonstrate the applicability and validity of the proposed method.
Originality/value
The numerical method is purposed for solving Hadamard-type fractional differential equations.
Details
Keywords
Zain ul Abdeen and Mujeeb ur Rehman
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Abstract
Purpose
The purpose of this paper is to present a computational technique based on Newton–Cotes quadrature rule for solving fractional order differential equation.
Design/methodology/approach
The numerical method reduces initial value problem into a system of algebraic equations. The method presented here is also applicable to non-linear differential equations. To deal with non-linear equations, a recursive sequence of approximations is developed using quasi-linearization technique.
Findings
The method is tested on several benchmark problems from the literature. Comparison shows the supremacy of proposed method in terms of robust accuracy and swift convergence. Method can work on several similar types of problems.
Originality/value
It has been demonstrated that many physical systems are modelled more accurately by fractional differential equations rather than classical differential equations. Therefore, it is vital to propose some efficient numerical method. The computational technique presented in this paper is based on Newton–Cotes quadrature rule and quasi-linearization. The key feature of the method is that it works efficiently for non-linear problems.