A NUMERICAL SCHEME FOR THE SOLUTION OF $q$-FRACTIONAL DIFFERENTIAL EQUATION USING $q$-LAGUERRE OPERATIONAL MATRIX

2020 ◽  
Vol 10 (1) ◽  
pp. 145-154
Author(s):  
B. Madhavi
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Numerical Scheme ◽  
Operational Matrix ◽  
Fractional Differential

scholarly journals Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Xin Liu ◽  
Kamran ◽  
Yukun Yao
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Numerical Scheme ◽  
Robust Stabilization ◽  
Smooth Curve ◽  
Quadrature Rule ◽  
Network Synthesis ◽  
Riccati Differential Equation ◽  
Bromwich Integral ◽  
Fractional Differential

The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order 0<β<1 with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.

scholarly journals Numerical solution of fractional differential equation by wavelets and hybrid functions

2018 ◽  
Vol 36 (2) ◽  
pp. 231 ◽  
Author(s):  
Amir Hosein Refahi Sheikhani ◽  
Mahamad Mashoof
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Legendre Polynomials ◽  
Linear Equations ◽  
Operational Matrix ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Fractional Differential

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By uses these methods we translate an FDE to an algebraic linear equations which can be solve. Methods has been tested by some numerical examples.

scholarly journals An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Jianping Liu ◽  
Xia Li ◽  
Limeng Wu
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Chebyshev Polynomial ◽  
Algebraic System ◽  
Operational Matrix ◽  
Main Idea ◽  
Operational Matrices ◽  
Variable Order ◽  
Fractional Differential ◽  
Variable Order Fractional Derivative

The multiterm fractional differential equation has a wide application in engineering problems. Therefore, we propose a method to solve multiterm variable order fractional differential equation based on the second kind of Chebyshev Polynomial. The main idea of this method is that we derive a kind of operational matrix of variable order fractional derivative for the second kind of Chebyshev Polynomial. With the operational matrices, the equation is transformed into the products of several dependent matrices, which can also be viewed as an algebraic system by making use of the collocation points. By solving the algebraic system, the numerical solution of original equation is acquired. Numerical examples show that only a small number of the second kinds of Chebyshev Polynomials are needed to obtain a satisfactory result, which demonstrates the validity of this method.

scholarly journals The generalized Gegenbauer-Humberts wavelet for solving fractional differential equations

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 107-118
Author(s):  
Jumana Alkhalissi ◽  
Ibrahim Emiroglu ◽  
Aydin Secer ◽  
Mustafa Bayram
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
New Method ◽  
Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
Operational Matrix Of Integration

In this paper we present a new method of wavelets, based on generalized Gegen?bauer-Humberts polynomials, named generalized Gegenbauer-Humberts wave?lets. The operational matrix of integration are derived. By using the proposed method converted linear and non-linear fractional differential equation a system of algebraic equations. In addition, discussed some examples to explain the efficiency and accuracy of the presented method.

scholarly journals The generalized Gegenbauer-Humberts wavelet for solving fractional differential equations

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 107-118
Author(s):  
Jumana Alkhalissi ◽  
Ibrahim Emiroglu ◽  
Aydin Secer ◽  
Mustafa Bayram
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
New Method ◽  
Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
Operational Matrix Of Integration

In this paper we present a new method of wavelets, based on generalized Gegen?bauer-Humberts polynomials, named generalized Gegenbauer-Humberts wave?lets. The operational matrix of integration are derived. By using the proposed method converted linear and non-linear fractional differential equation a system of algebraic equations. In addition, discussed some examples to explain the efficiency and accuracy of the presented method.

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