scholarly journals Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

2019 ◽  
Vol 24 (2) ◽  
pp. 176-188 ◽  
Author(s):  
Eid H. Doha ◽  
Mohamed A. Abdelkawy ◽  
Ahmed Z.M. Amin ◽  
Dumitru Baleanu
Keyword(s):  
Chebyshev Polynomials ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Riccati Differential Equations ◽  
Spectral Technique ◽  
Expansion Coefficients

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

scholarly journals Numerical Algorithm to Solve a Class of Variable Order Fractional Integral-Differential Equation Based on Chebyshev Polynomials

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Kangwen Sun ◽  
Ming Zhu
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Numerical Algorithm ◽  
Chebyshev Polynomials ◽  
Fractional Integral ◽  
Numerical Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Integral Differential Equation

The purpose of this paper is to study the Chebyshev polynomials for the solution of a class of variable order fractional integral-differential equation. The properties of Chebyshev polynomials together with the four kinds of operational matrixes of Chebyshev polynomials are used to reduce the problem to the solution of a system of algebraic equations. By solving the algebraic equations, the numerical solutions are acquired. Further some numerical examples are shown to illustrate the accuracy and reliability of the proposed approach and the results have been compared with the exact solution.

scholarly journals Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals New Numerical Approach for Solving Abel’s Integral Equations

2021 ◽  
Vol 46 (3) ◽  
pp. 255-271
Author(s):  
Ayşe Anapalı Şenel ◽  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Efficient Method ◽  
Fractional Derivatives ◽  
Numerical Approach ◽  
Main Character ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Taylor Expansions

Abstract In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

scholarly journals New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
MohammadHossein Derakhshan
Keyword(s):  
Fractional Derivatives ◽  
Numerical Approximation ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Integrodifferential Equations ◽  
Variable Order ◽  
Algebraic Equations ◽  
Lagrange Multiplier Technique ◽  
Derivatives Of ◽  
Variable Order Fractional Derivative

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

scholarly journals Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
H. Jafari ◽  
S. Nemati ◽  
R. M. Ganji
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Original Problem ◽  
High Accuracy ◽  
Operational Matrices ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Tests ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations

AbstractIn this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

A Numerical Approach for Fractional Order Riccati Differential Equation Using B-Spline Operational Matrix

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Suggested Approach ◽  
B Spline ◽  
Riccati Differential Equations ◽  
The Given

AbstractIn this article, we develop an effective numerical method to achieve the numerical solutions of nonlinear fractional Riccati differential equations. We found the operational matrix within the linear B-spline functions. By this technique, the given problem converts to a system of algebraic equations. This technique is used to solve fractional Riccati differential equation. The obtained results are illustrated both applicability and validity of the suggested approach.

scholarly journals New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Main Idea ◽  
Initial Condition ◽  
Algebraic Equations ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Expansion Coefficients

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented.

Sign in / Sign up

Export Citation Format

Share Document