Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad

On the Solution of Volterra Integro-differential Equations using a Modified Adomian Decomposition Method

The Adomian decomposition method’s effectiveness has been demonstrated in recent research, the process requires several iterations and can be time-consuming. By breaking down the source term function into series, the current work introduced a new decomposition approach to the Adomian decomposition method. As compared to the conventional Adomian decomposition approach, the newly devised method hastens the convergence of the solution. Numerical experiments were provided to show the superiority qualities

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations

Multiscale wavelet analysis for integral and differential problems

2009 - 2010The object of the present research is wavelet analysis of integral and differential problems by means of harmonic and circular wavelets. It is shown that circular wavelets constitute a complete basis for L2[0; 1] functions, and form multiresolution analysis. Multiresolution analysis can be briefly considered as a decomposition of L2[0; 1] into a complete set of scale depending subspaces of wavelets. Thus, integral operators, differential operators, and L2(R) functions were investigated as scale depending functions through their projection onto these subspaces of wavelets. In particular: - conditions when a certain wavelet can be applied for solution of integral or differential problem are given; - it is shown that the accuracy of this approach exponentially grows when increasing the number of vanishing moments and scaling parameter; - wavelet solutions of low-dimensional nonlinear partial differential equations are compared with other methods; - wavelet-based approach is applied to low-dimensional Fredholm integral equations and the Galerkin method for two-dimensional Fredholm integral equations.[edited by author]. Oggetto della seguente ricerca `e l’analisi di problemi differenziali e integrali, utilizzando wavelet armoniche e wavelet armoniche periodiche. Si dimostra che le wavelet periodiche costituiscono una base completa per le funzioni L2[0; 1] e formano un’analisi multiscala. L’analisi multirisoluzione pu`o essere brevemente considerata come la decomposizione di L2[0; 1] in un insieme completo di sottospazi di wavelet dipendenti da un fattore di scala. Pertanto gli operatori integrali e differenziali e le funzioni L2(R) vengono studiati come funzioni di scala mediante le corrispondenti proiezioni in questi sottospazi di wavelet. In particolare, vengono sviluppati quattro principali argomenti: - sono state individuate le condizioni per applicare una data famiglia di wavelets alla soluzione di un data problema differenziale o integrale; - si `e dimostrato che la precisione di questo approccio cresce esponenzialmente quando decresce il numero dei momenti nulli e del parametro di scala; - soluzioni wavelet di equazioni differenziali a derivate parziali nonlineari di dimensione bassa sono state confrontate con altri metodi di soluzioni; - l’approccio basato sull’uso delle wavelet `e stato applicato anche per ricerca di soluzioni di alcune equazioni integrali di Fredholm e insieme al metodo di Galerkin per risolvere equazioni integrali Fredholm di dimensioni due.[a cura dell'autore]IX n.s

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms

Effective numerical methods for nonlinear singular two-point boundary value Fredholm integro-differential equations

We deal with some effective numerical methods for solving a class of nonlinear singular two-point boundary value Fredholm integro-differential equations. Using an appropriate interpolation and a q-order quadrature rule of integration, the original problem will be approximated by the non-linear finite difference equations and so reduced to a nonlinear algebraic system that can be simply implemented. The convergence properties of the proposed method are discussed, and it is proved that its convergence order will be of O(hmin{ 72 ,q− 12 }). Ample numerical results are addressed to con-firm the expected convergence order as well as the accuracy and efficiency of the proposed method

On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem

In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a,\ua0b] or [0,\ua01]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of a (or origin) from the interval, where we only take L2-norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson(Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples