用修正的伯努利多项式求解线性积分微分方程的新方法
A new technique to solve linear integro-differential equations (IDEs) with modified Bernoulli polynomials
摘要：
本文提出了一种求线性积分微分方程近似解的新方法。该方法基于修正的标准正交伯努利多项式及其运算矩阵。该方法将一个给定的积分微分方程转化为一组系数未知的代数方程组，借助于方程中出现的已知函数、修正的伯努利多项式和运算矩阵，很容易得到一组系数未知的代数方程组。以所需次数的多项式的形式得到了近似解。将该方法应用于三个著名的积分微分方程，以证明该方法的准确性和有效性。给出了近似解的数值结果，并与现有的精确解进行了比较。通过数值比较观察到相当小的近似误差，这是进一步减少到所需的显着性水平。该方法比现有的许多方法都要简单和简短。
英文摘要：
In this work, a new technique has been presented to find approximate solution of linear integro-differential equations. The method is based on modified orthonormal Bernoulli polynomials and an operational matrix thereof. The method converts a given integro-differential equation into a set of algebraic equations with unknown coefficients, which is easily obtained with help of the known functions appearing in the equation, modified Bernoulli polynomials and operational matrix. Approximate solution is obtained in form of a polynomial of required degree. The method is also applied to three well known integro-differential equations to demonstrated the accuracy and efficacy of the method. Numerical results of approximate solution are plotted to compared with available exact solutions. Considerably small error of approximation is observed through numerical comparison, which is further reducible to a required level of significance. Method is comparatively simpler and shorter than many existing methods.
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