scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

scholarly journals THE METHOD OF NUMERICAL SOLUTION OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND

2018 ◽  
pp. 10-18
Author(s):  
Karakeev T.T. ◽  
Mustafayeva N.T.
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Initial Point ◽  
Nonlinear Problems ◽  
Volterra Integral Equations ◽  
Algebraic Equations

When considering systems of differential equations with very general boundary conditions, exact solution methods encounter great difficulties, which become insurmountable in the study of nonlinear problems. In this case it is necessary to apply to certain numerical methods. It is important to note that the use of numerical methods often allows you to abandon the simplified interpretation of the mathematical model of the process. The problems of numerical solution of nonlinear Volterra integral equations of the first kind with a differentiable kernel, which degenerates at the initial point of the diagonal, are studied in the paper. This equation is reduced to the Volterra integral equation of the third kind and a numerical method is developed on the basis of that regularized equation. The convergence of the numerical solution to the exact solution of the Volterra integral equation of the first kind is proved, an estimate of the permissible error and a recursive formula of the computational process are obtained. Keywords: nonlinear integral equation, system of nonlinear algebraic equations, error vectors, the Volterra equation, small parameter, numerical methods.

Variational Iteration Method for a Nonlinear Reaction-Diffusion Process

2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Shu-Qiang Wang ◽  
Ji-Huan He
Keyword(s):  
Exact Solution ◽  
Diffusion Process ◽  
Iteration Method ◽  
Unknown Parameter ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Rigorous Derivation ◽  
Variational Iteration ◽  
Nonlinear Reaction ◽  
The Given

An extremely simple and elementary, but rigorous derivation of temperature distribution of a reaction-diffusion process is given using the variational iteration method. In this method, a trial function (an initial solution) is chosen with some unknown parameter, which is identified after a few iterations according to the given boundary conditions. Comparison with the exact solution shows that the method is very effective and convenient.

scholarly journals Numerical methods for a Volterra integral equation with non-smooth solutions

2006 ◽  
Vol 189 (1-2) ◽  
pp. 412-423 ◽  
Author(s):  
Teresa Diogo ◽  
Neville J. Ford ◽  
Pedro Lima ◽  
Svilen Valtchev
Keyword(s):  
Integral Equation ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Smooth Solutions

scholarly journals The Approximate Solution of Fractional Fredholm Integrodifferential Equations by Variational Iteration and Homotopy Perturbation Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Variational Iteration Method ◽  
Integrodifferential Equations ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Constant Coefficients

Variational iteration method and homotopy perturbation method are used to solve the fractional Fredholm integrodifferential equations with constant coefficients. The obtained results indicate that the method is efficient and also accurate.

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