scholarly journals Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-30 ◽  
Author(s):  
Fayyaz Ahmad ◽  
Shafiq Ur Rehman ◽  
Malik Zaka Ullah ◽  
Hani Moaiteq Aljahdali ◽  
Shahid Ahmad ◽  
...  
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Lu Factorization ◽  
Single Application ◽  
Initial Value ◽  
Comprehensive Comparison

In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is 5m-11, where m is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.

scholarly journals An efficient 4-step block method for solution of first order initial value problems via shifted chebyshev polynomial

2021 ◽  
Vol 1 (2) ◽  
pp. 25-36
Author(s):  
Isah O. ◽  
Salawu S. ◽  
Olayemi S. ◽  
Enesi O.
Keyword(s):  
Exact Solution ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Test Problems ◽  
Block Method ◽  
Initial Value ◽  
Order Zero ◽  
First Order ◽  
Continuous Scheme ◽  
Shifted Chebyshev Polynomial

In this paper, we develop a four-step block method for solution of first order initial value problems of ordinary differential equations. The collocation and interpolation approach is adopted to obtain a continuous scheme for the derived method via Shifted Chebyshev Polynomials, truncated after sufficient terms. The properties of the proposed scheme such as order, zero-stability, consistency and convergence are also investigated. The derived scheme is implemented to obtain numerical solutions of some test problems, the result shows that the new scheme competes favorably with exact solution and some existing methods.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

scholarly journals Note on the Numerical Solutions of the General Matrix Convolution Equations by Using the Iterative Methods and Box Convolution Product

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Adem Kılıçman ◽  
Zeyad Al zhour
Keyword(s):  
Exact Solutions ◽  
Iterative Methods ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Convolution Product ◽  
Initial Value ◽  
General Matrix ◽  
Convolution Equations

We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these solutions consistently converge to the exact solutions and independent of the initial value.

scholarly journals On the iterative methods for solving fractional initial value problems: new perspective

2021 ◽  
Vol 2 (1) ◽  
pp. 76-81
Author(s):  
Qasem M. Al-Mdallal ◽  
Mohamed Ali Hajji ◽  
Thabet Abdeljawad
Keyword(s):  
Approximate Solution ◽  
Numerical Solution ◽  
Iterative Methods ◽  
Initial Value Problems ◽  
Short Communication ◽  
Initial Value ◽  
New Perspective

In this short communication, we introduce a new perspective for a numerical solution of fractional initial value problems (FIVPs). Basically, we split the considered FIVP into FIVPs on subdomains which can be solved iteratively to obtain the approximate solution for the whole domain.

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