234 research outputs found
NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD
In this study, Akbari-Ganji’s Method (AGM) was applied to solve Volterra Integro-Differential Difference Equations (VIDDE) using Legendre polynomials as basis functions. Here, a trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. The unknown coefficients are solved for using the new proposed approach, AGM which principally involves the application of the boundary conditions on successive derivatives and integrals of the problem to obtain a system of equations. The system of equation is solved using any appropriate computer software, Maple 18. Some examples were solved and the results compared to the exact solutions
Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations
In this paper, we use a numerical method that involves hybrid and block-pulse
functions to approximate solutions of systems of a class of Fredholm and
Volterra integro-differential equations. The key point is to derive a new
approximation for the derivatives of the solutions and then reduce the
integro-differential equation to a system of algebraic equations that can be
solved using classical methods. Some numerical examples are dedicated for
showing efficiency and validity of the method that we introduce
An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions
This paper is concerned with the numerical solution of the third kind
Volterra integral equations with non-smooth solutions based on the recursive
approach of the spectral Tau method. To this end, a new set of the fractional
version of canonical basis polynomials (called FC-polynomials) is introduced.
The approximate polynomial solution (called Tau-solution) is expressed in terms
of FC-polynomials. The fractional structure of Tau-solution allows recovering
the standard degree of accuracy of spectral methods even in the case of
non-smooth solutions. The convergence analysis of the method is studied. The
obtained numerical results show the accuracy and efficiency of the method
compared to other existing methods
A M\"untz-Collocation spectral method for weakly singular volterra integral equations
In this paper we propose and analyze a fractional Jacobi-collocation spectral
method for the second kind Volterra integral equations (VIEs) with weakly
singular kernel . First we develop a family of fractional
Jacobi polynomials, along with basic approximation results for some weighted
projection and interpolation operators defined in suitable weighted Sobolev
spaces. Then we construct an efficient fractional Jacobi-collocation spectral
method for the VIEs using the zeros of the new developed fractional Jacobi
polynomial. A detailed convergence analysis is carried out to derive error
estimates of the numerical solution in both - and weighted
-norms. The main novelty of the paper is that the proposed method is
highly efficient for typical solutions that VIEs usually possess. Precisely, it
is proved that the exponential convergence rate can be achieved for solutions
which are smooth after the variable change for a
suitable real number . Finally a series of numerical examples are
presented to demonstrate the efficiency of the method
Multiple Perturbed Collocation Tau Method for Solving Nonlinear Integro-Differential Equations
The purpose of the study was to investigate the numerical solution of non-linear Fredholm and Volterra integro-differential equations by the proposed method called Multiple Perturbed Collocation Tau Method (MPCTM). We assumed a perturbed approximate solution in terms of Chebyshev polynomial basis function and then determined the derivatives of the perturbed approximate solution which are then substituted into the special classes of the problems considered. Thus, resulting into n-folds integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Newton’s method which are then substituted back into the approximate solution.Illustrative examples are given to demonstrate the efficiency, computational cost and accuracy of the method. The results obtained with some numerical examples are compared favorable with some existing numerical methods in literature and with the exact solutions where they are known in closed form.Keywords: Nonlinear Problems, Tau Method, Integro-Differential, Newton’s method
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