scholarly journals A New Tau Method for Solving Nonlinear Lane-Emden Type Equations via Bernoulli Operational Matrix of Differentiation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
E. Tohidi ◽  
Kh. Erfani ◽  
M. Gachpazan ◽  
S. Shateyi
Keyword(s):  
Error Analysis ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Bernoulli Polynomial ◽  
Tau Method ◽  
Algebraic Equations ◽  
Fundamental Structure ◽  
Polynomial Approximations ◽  
Mild Conditions ◽  
Operational Matrix Of Differentiation

A new and efficient numerical approach is developed for solving nonlinear Lane-Emden type equations via Bernoulli operational matrix of differentiation. The fundamental structure of the presented method is based on the Tau method together with the Bernoulli polynomial approximations in which a new operational matrix is introduced. After implementation of our scheme, the main problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefficients. Also, under several mild conditions the error analysis of the proposed method is provided. Several examples are included to illustrate the efficiency and accuracy of the proposed technique and also the results are compared with the different methods. All calculations are done in Maple 13.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

scholarly journals Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

2021 ◽  
Vol 5 (3) ◽  
pp. 100
Author(s):  
Youssri Hassan Youssri
Keyword(s):  
Differential Equation ◽  
Operational Matrix ◽  
Initial Guess ◽  
Test Problems ◽  
Tau Method ◽  
Algebraic Equations ◽  
Differential Problem ◽  
Riccati Differential Equation ◽  
Ultraspherical Polynomials ◽  
Expansion Coefficients

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

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 A numerical approach based on $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials for solving fractional-order cohomological equations with variable coefficients

2021 ◽  
Author(s):  
Akbar Dehghan Nezhad ◽  
Mina Moghaddam Zeabadi
Keyword(s):  
Fractional Order ◽  
Continuous Time ◽  
Legendre Polynomials ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Variable Coefficients ◽  
Algebraic Equations ◽  
Cohomological Equation ◽  
Liouville Derivative ◽  
Cohomological Equations

This research presents a numerical approach to obtain the approximate solution of the n-dimensional cohomological equations of fractional order in continuous-time dynamical systems. For this purpose, the $ n $-dimensional fractional M\”{u}ntz-Legendre polynomials (or n-DFMLPs) are introduced. The operational matrix of the fractional Riemann-Liouville derivative is constructed by employing n-DFMLPs. Our method transforms the cohomological equation of fractional order into a system of algebraic equations. Therefore, the solution of that system of algebraic equations is the solution of the associated cohomological equation. The error bound and convergence analysis of the applied method under the $ L^{2} $-norm is discussed. Some examples are considered and discussed to confirm the efficiency and accuracy of our method.

scholarly journals New method for solving a class of fractional partial differential equations with applications

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 277-286 ◽  
Author(s):  
Hossein Jafari ◽  
Haleh Tajadodi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Linear Algebraic Equations ◽  
Spline Polynomial

In this work we suggest a numerical approach based on the B-spline polynomial to obtain the solution of linear fractional partial differential equations. We find the operational matrix for fractional integration and then we convert the main problem into a system of linear algebraic equations by using this matrix. Examples are provided to show the simplicity of our method.

scholarly journals NUMERICAL SOLUTION OF MIXED LINEAR VOLTERRA-FREDHOLM INTEGRAL EQUATIONS BY MODIFIED BLOCK PULSE FUNCTIONS

2021 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Linear System ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration ◽  
Modified Block

A numerical method based on modified block pulse functions is proposed for solving the mixed linear Volterra-Fredholm integral equations. We obtain an integration operational matrix of modified block pulse functions on interval [0,T). A modified block pulse functions and their operational matrix of integration, the mixed linear Volterra-Fredholm integral equations can be reduced to a linear system of algebraic equations. The rate of convergence is O(h) and error analysis of the proposed method are discussed. Some examples are provided to show that the proposed method have a good degree of accuracy.

scholarly journals An Efficient Spectral Approximation for Solving Several Types of Parabolic PDEs with Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
E. Tohidi ◽  
A. Kılıçman
Keyword(s):  
Boundary Conditions ◽  
Operational Matrix ◽  
Nonlocal Boundary ◽  
Main Idea ◽  
Nonlocal Boundary Conditions ◽  
Algebraic Equations ◽  
Parabolic Pdes ◽  
Direct Collocation ◽  
Operational Matrix Of Differentiation ◽  
The Given

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy.

scholarly journals The Solutions of Second Order Nonlinear Two Point Boundary Value Problems

2019 ◽  
Vol 4 (8) ◽  
pp. 49-54
Author(s):  
Abdurkadir Edeo Gemeda
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuation Method ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Second Order ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Operational Matrix Of Differentiation

In this paper, generalized shifted Legendre polynomial approximation on a given arbitrary interval has been designed to find an approximate solution of a given second order nonlinear two point boundary value problems of ordinary differential equations. Here an approach using Tau method based on Legendre operational matrix of differentiation [2] & [5] has been addressed to generate the nonlinear systems of algebraic equations. The unknown Legendre coefficients of these nonlinear systems are the solutions of the system and they have been solved by continuation method. These unknown Legendre coefficients are then used to write the approximate solutions to the second order nonlinear two point boundary value problems. The validity and efficiency of the method has also been illustrated with numerical examples and graphs assisted by MATLAB.

scholarly journals A new numerical approach for solving high-order linear and non-linear differantial equations

2018 ◽  
Vol 22 (Suppl. 1) ◽  
pp. 67-77
Author(s):  
Aydin Secer ◽  
Selvi Altun
Keyword(s):  
Matrix Method ◽  
Operational Matrix ◽  
Numerical Approach ◽  
High Order ◽  
Algebraic Equations ◽  
Practical Application ◽  
Order Linear ◽  
Operational Matrix Method ◽  
Non Linear ◽  
Linear Problems

In this paper, the Legendre wavelet operational matrix method has been introduced for solving high-order linear and non-linear multi-point: initial and boundary value problems. It has been suggested that the technique is rest upon practical application of the operational matrix and its derivatives. The differential equation is presented that it is converted to a system of algebraic equations via the properties of Legendre wavelet together with the operational matrix method. As a result of this study, the scheme has been tested on five linear and non-linear problems. The results have demonstrated that this method is a very effective and advantageous tool in solving such problems.

scholarly journals Numerical Solution of Fuzzy Fractional Pharmacokinetics Model Arising from Drug Assimilation into the Bloodstream

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Ali Ahmadian ◽  
Norazak Senu ◽  
Farhad Larki ◽  
Soheil Salahshour ◽  
Mohamed Suleiman ◽  
...  
Keyword(s):  
Error Analysis ◽  
Fractional Derivative ◽  
Numerical Solution ◽  
Error Estimation ◽  
Upper Bound ◽  
Tau Method ◽  
Algebraic Equations ◽  
Special Cases ◽  
The Absolute ◽  
Fractional Pharmacokinetics

We propose a Jacobi tau method for solving a fuzzy fractional pharmacokinetics. This problem can model the concentration of the drug in the blood as time increases. The proposed approach is based on the Jacobi tau (JT) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The method reduces the solution of the problem to the solution of a system of algebraic equations. Error analysis included the fractional derivative error estimation, and the upper bound of the absolute errors is introduced for this method.

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