Numerical solution of nonlinear algebraic equations in stiff ODE solving (1986--89)---Quasi-Newton updating for large scale nonlinear systems (1989--90)

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.

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Numerical Solution of Nonlinear Systems of Algebraic Equations

International Journal of Data Science and Analysis ◽

10.11648/j.ijdsa.20180401.14 ◽

2018 ◽

Vol 4 (1) ◽

pp. 20

Author(s):

Kamoh Nathaniel Mahwash

Keyword(s):

Nonlinear Systems ◽

Numerical Solution ◽

Algebraic Equations

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Numerical Solution of Fractional Sine-Gordon Equation Using Spectral Method and Homogenization

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2018-0339 ◽

2019 ◽

Vol 20 (7-8) ◽

pp. 811-822

Author(s):

Maryam Hasanpour ◽

Mahmoud Behroozifar ◽

Nazanin Tafakhori

Keyword(s):

Numerical Solution ◽

Jacobi Polynomials ◽

Second Step ◽

Test Problems ◽

Algebraic Equations ◽

Newton’S Iterative Method ◽

Main Equation ◽

Dimensional Approximation ◽

Nonlinear Algebraic Equations ◽

Sg Equation

AbstractIn this paper, a new method for the numerical solution of fractional sine-Gordon (SG) equation is presented. Our method consists of two steps, in first step: the main equation is converted to a homogeneous one using interpolation. In second step: two-dimensional approximation of functions by shifted Jacobi polynomials is used to reduce the problem into a system of nonlinear algebraic equations. The archived system is solved by Newton’s iterative method. Our method is stated in general case on rectangular [a,b] × [0,T] which is based upon Jacobi polynomial by parameters (α,β). Several test problems are employed and results of numerical experiments are presented and also compared with analytical solutions. Also, we verify the numerical stability of the method, by applying a disturbance in the problem. The obtained results confirm the acceptable accuracy and stability of the presented method.

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Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

Abstract and Applied Analysis ◽

10.1155/2013/416757 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

S. H. Behiry

Keyword(s):

Numerical Solution ◽

Numerical Method ◽

Present Method ◽

Bernstein Polynomials ◽

Integrodifferential Equations ◽

Algebraic Equations ◽

Numerical Examples ◽

Hybrid Functions ◽

Block Pulse Functions ◽

Nonlinear Algebraic Equations

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

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