حل عددی معادلات انتگرال –دیفرانسیل تاخیری فردهلم خطی مرتبۀ بالا با ضرایب متغیر با استفاده از بسط چبیشف

ایدۀ اصلی این مقاله، استفاده از چندجمله‌ای‌های چبیشف برای حل معادلات انتگرال-دیفرانسیل تاخیری فردهلم خطی با مراتب بالا است. معمولاً حل این معادلات به‌روش‌های تحلیلی امکان‌پذیر نیست یا در صورت امکان بسیار مشکل است. در این روش معادله مورد نظر به‌وسیلۀ روابط ماتریسی بین چندجمله‌ای‌های چبیشف و مشتقات آنها به دستگاه معادلات خطی تبدیل می‌شود. ماتریس‌های عملیاتی عملگرهای تاخیر و مشتق همراه با روش تائو برای محاسبۀ ضرایب مجهول بسط چبیشف جواب استفاده می‌شوند. همگرایی روش بررسی شده است. مثال‌های عددی، اعتبار و کارایی روش ارائه شده را نشان می‌دهند. هم‌چنین نتایج حاصل از روش با نتایج موجود مقایسه شده است.

Numerical Solution using Chebyshev Expansion of the Higher-Orders Linear Fredholm Integro-Differential-Difference Equations with Variable Coefficients

The main aim of this paper is to apply the Chebyshev polynomials for the solution of the linear Fredholm integrodifferentialdifference equation of high orders. It is usually difficult to analytically solve this equation. Our approach consists of reducing the problem to a set of linear equations by means of the matrix relations between the Chebyshev polynomials and their derivatives. The operational matrices of delay and derivative together with the Tau method are then utilized to evaluate the unknown coefficients of Chebyshev expansion of the solution. The convergence analysis is studied. Illustrative examples show the validity and applicability of the presented technique. Also, a comparison is made with existing results. IntroductionThe integrodifference equations arise in different applications such as biological, physical and engineering problems. In recent years, there has been a growing interest in the numerical treatment of the integrodifferentialdifference equations. Since the mentioned equations are usually difficult to solve analytically, numerical methods are required. Several numerical methods were used such as successive approximation method, Adomian decomposition method, the Taylor collocation method, Haar wavelet method, Legendre wavelets method, waveletGalerkin method, monotone iterative technique, Walsh series method, etc.In this work, we develop a framework to obtain the numerical solution of the sorder linear Fredholm integrodifferentialdifference equation with variable coefficients.under the mixed conditionswhereand are known continuous functions. Here, the real coefficients and are given constants.Our approach consists of reducing the problem to a set of linear equations by expanding the solution in terms of Chebyshev polynomials. The operational matrices of delay and derivative are given. These matrices together with the Tau method utilized to evaluate the unknown coefficients of expansion. The Tau method has been originally proposed by Lanczos for ordinary differential equations and extended by Ortiz. The method consists of expanding the required approximate solution as the elements of a complete set of orthogonal polynomials. Recently there have been several published works in the literature on the applications of the Tau method.ConclusionThis paper deals with the solution of linear Fredholm integrodifferentialdifference equations of high order with variable coefficients. Our approach was based on the Chebyshev Tau method which reduces a linear Fredholm integrodifferentialdifference equation into a set of linear algebraic equations. Numerical results show that this approach can solve the problem effectively. The approach, with some modifications, can be employed to solve differentialdifference equations and Fredholm integrodifferential equations../files/site1/files/64/4.pdf