scholarly journals A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

2013 ◽  
Vol 37 (6) ◽  
pp. 4283-4294 ◽  
Author(s):  
E. Tohidi ◽  
A.H. Bhrawy ◽  
K. Erfani
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Pantograph Equation

scholarly journals Numerical solution of two-dimensional nonlinear fractional order reaction-advection-diffusion equation by using collocation method

2021 ◽  
Vol 29 (2) ◽  
pp. 211-230
Author(s):  
Manpal Singh ◽  
S. Das ◽  
Rajeev ◽  
E-M. Craciun
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Existing Problems ◽  
Advection Diffusion Equation ◽  
Error Analyses ◽  
Nonlinear Algebraic Equations ◽  
Accurate Numerical Solution

Abstract In this article, two-dimensional nonlinear and multi-term time fractional diffusion equations are solved numerically by collocation method, which is used with the help of Lucas operational matrix. In the proposed method solutions of the problems are expressed in terms of Lucas polynomial as basis function. To determine the unknowns, the residual, initial and boundary conditions are collocated at the chosen points, which produce a system of nonlinear algebraic equations those have been solved numerically. The concerned method provides the highly accurate numerical solution. The accuracy of the approximate solution of the problem can be increased by expanding the terms of the polynomial. The accuracy and efficiency of the concerned method have been authenticated through the error analyses with some existing problems whose solutions are already known.

scholarly journals Numerical solution of time-fractional telegraph equation by using a new class of orthogonal polynomials

2022 ◽  
Vol 40 ◽  
pp. 1-13
Author(s):  
Fakhrodin Mohammadi ◽  
Hossein Hassani
Keyword(s):  
Approximate Solution ◽  
Error Analysis ◽  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Telegraph Equation ◽  
New Class ◽  
Fractional Telegraph Equation ◽  
Collocation Approach ◽  
Chelyshkov Polynomials

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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