Spectral Solutions for Multi-Term Fractional Initial Value Problems Using a New Fibonacci Operational Matrix of Fractional Integration

2016 ◽  
Vol 2 (2) ◽  
pp. 141-151 ◽  
Author(s):  
Youssri H. Youssri ◽  
Waleed M. Abd-Elhameed
Keyword(s):  
Initial Value Problems ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Initial Value

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method

2021 ◽  
Vol 2 (2) ◽  
pp. 57-67
Author(s):  
Bushra Esaa Kashem ◽  
Suha SHIHAB
Keyword(s):  
Approximate Solution ◽  
Initial Value Problems ◽  
Operational Matrix ◽  
Mathematical Physics ◽  
Initial Value ◽  
Solution Function ◽  
Emden Equation ◽  
Physical Problems ◽  
Operational Matrix Of Integration ◽  
The Given

Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations.

scholarly journals A decomposition algorithm coupled with operational matrices approach with applications to fractional differential equations

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 449-455
Author(s):  
Imran Talib ◽  
Nur Alam ◽  
Dumitru Baleanu ◽  
Danish Zaidi
Keyword(s):  
Numerical Method ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Legendre Function ◽  
Operational Matrix ◽  
Decomposition Algorithm ◽  
Operational Matrices ◽  
Initial Value ◽  
Multiple Orders ◽  
Fractional Differential

In this article, we solve numerically the linear and non-linear fractional initial value problems of multiple orders by developing a numerical method that is based on the decomposition algorithm coupled with the operational matrices approach. By means of this, the fractional initial value problems of multiple orders are decomposed into a system of fractional initial value problems which are then solved by using the operational matrices approach. The efficiency and advantage of the developed numerical method are highlighted by comparing the results obtained otherwise in the literature. The construction of the new derivative operational matrix of fractional legendre function vectors in the Caputo sense is also a part of this research. As applications, we solve several fractional initial value problems of multiple orders. The numerical results are displayed in tables and plots.

scholarly journals Solving Higher-Order Boundary and Initial Value Problems via Chebyshev–Spectral Method: Application in Elastic Foundation

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 987 ◽  
Author(s):  
Praveen Agarwal ◽  
Maryam Attary ◽  
Mohammad Maghasedi ◽  
Poom Kumam
Keyword(s):  
Elastic Foundation ◽  
Initial Value Problems ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Operational Matrix ◽  
Engineering Problem ◽  
Initial Value ◽  
Equivalent Representation ◽  
Efficient Scheme ◽  
Sufficient Precision

In this work, we introduce an efficient scheme for the numerical solution of some Boundary and Initial Value Problems (BVPs-IVPs). By using an operational matrix, which was obtained from the first kind of Chebyshev polynomials, we construct the algebraic equivalent representation of the problem. We will show that this representation of BVPs and IVPs can be represented by a sparse matrix with sufficient precision. Sparse matrices that store data containing a large number of zero-valued elements have several advantages, such as saving a significant amount of memory and speeding up the processing of that data. In addition, we provide the convergence analysis and the error estimation of the suggested scheme. Finally, some numerical results are utilized to demonstrate the validity and applicability of the proposed technique, and also the presented algorithm is applied to solve an engineering problem which is used in a beam on elastic foundation.

scholarly journals A Novel Spectral Modified Pell Polynomials for Solving Singular Differential Equations

2021 ◽  
Vol 32 (1) ◽  
pp. 18
Author(s):  
Mohammed Abdelhadi Sarhan ◽  
Suha SHIHAB ◽  
Mohammed RASHEED
Keyword(s):  
Initial Value Problems ◽  
Operational Matrix ◽  
Exact Formula ◽  
Computational Method ◽  
Initial Value ◽  
Spectral Techniques ◽  
Expansion Coefficients ◽  
New Formula ◽  
Expansion Scheme

This paper studies the modified Pell polynomials. Some important properties of modified Pell polynomials are presented. An exact formula of modified Pell polynomials derivative in terms of modified Pell themselves is first derived with the proof and then a new relationship is constructed which relates the modified Pell polynomials expansion coefficients of a derivative in terms of their original expansion coefficients. An interesting new formula for the product operational matrix of modified Pell polynomials is also derived in this work. With modified Pell polynomials expansion scheme, the powers 1, x, …, xn are expressed in terms of such polynomials. The main goal of all presented formulas is to simplify the original equations and the determination of the coefficients of expansion based on modified Pell polynomials will be easy. Spectral techniques together with all the derived formulas of modified Pell polynomials are utilized to solve some singular initial value problems. Three test examples are solved in this work to illustrate the validity of the proposed method. The computational method is replaced by exact and explicit formulas. More accurate results are obtained than those presented by other existing methods in the literature.

scholarly journals Chebyshev Operational Matrix Method for Lane-Emden Problem

2019 ◽  
Vol 8 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Bhuvnesh Sharma ◽  
Sunil Kumar ◽  
M.K. Paswan ◽  
Dindayal Mahato
Keyword(s):  
Chebyshev Polynomial ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Absolute Error ◽  
Initial Value ◽  
Operational Matrix Method ◽  
Modified Method ◽  
Operational Matrix Of Differentiation ◽  
Linear And Nonlinear

AbstractIn the this paper, a new modified method is proposed for solving linear and nonlinear Lane-Emden type equations using first kind Chebyshev operational matrix of differentiation. The properties of first kind Chebyshev polynomial and their shifted polynomial are first presented. These properties together with the operation matrix of differentiation of first kind Chebyshev polynomial are utilized to obtain numerical solutions of a class of linear and nonlinear LaneEmden type singular initial value problems (IVPs). The absolute error of this method is graphically presented. The proposed framework is different from other numerical methods and can be used in differential equations of the same type. Several examples are illuminated to reveal the accuracy and validity of the proposed method.

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