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首页> 外文期刊>American journal of applied sciences >Numerical Solution of Second-Order Linear Fredholm Integro-Differential Equation Using Generalized Minimal Residual Method
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Numerical Solution of Second-Order Linear Fredholm Integro-Differential Equation Using Generalized Minimal Residual Method

机译：二阶线性Fredholm积分微分方程的广义最小残值法数值解

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Problem statement: This research purposely brought up to solve complicated equations such as partial differential equations, integral equations, Integro-Differential Equations (IDE), stochastic equations and others. Many physical phenomena contain mathematical formulations such integro-differential equations which are arise in fluid dynamics, biological models and chemical kinetics. In fact, several formulations and numerical solutions of the linear Fredholm integro-differential equation of second order currently have been proposed. This study presented the numerical solution of the linear Fredholm integro-differential equation of second order discretized by using finite difference and trapezoidal methods. Approach: The linear Fredholm integro-differential equation of second order will be discretized by using finite difference and trapezoidal methods in order to derive an approximation equation. Later this approximation equation will be used to generate a dense linear system and solved by using the Generalized Minimal Residual (GMRES) method. Results: Several numerical experiments were conducted to examine the efficiency of GMRES method for solving linear system generated from the discretization of linear Fredholm integro-differential equation. For the comparison purpose, there are three parameters such as number of iterations, computational time and absolute error will be considered. Based on observation of numerical results, it can be seen that the number of iterations and computational time of GMRES have declined much faster than Gauss-Seidel (GS) method. Conclusion: The efficiency of GMRES based on the proposed discretization is superior as compared to GS iterative method.
机译：问题陈述：本研究旨在解决复杂的方程，例如偏微分方程，积分方程，积分微分方程（IDE），随机方程等。许多物理现象包含数学公式，例如积分微分方程，这些方程在流体动力学，生物学模型和化学动力学中产生。实际上，目前已经提出了线性二阶Fredholm积分-微分方程的几种公式和数值解。该研究提出了使用有限差分和梯形方法离散的二阶线性Fredholm积分-微分方程的数值解。方法：将使用有限差分和梯形方法离散二阶线性Fredholm积分微分方程，以得出一个近似方程。稍后，该近似方程将用于生成密集线性系统，并通过使用广义最小残差（GMRES）方法进行求解。结果：进行了一些数值实验，以检验GMRES方法求解线性Fredholm积分微分方程离散化产生的线性系统的效率。为了进行比较，将考虑三个参数，例如迭代次数，计算时间和绝对误差。根据数值结果的观察，可以看出GMRES的迭代次数和计算时间的下降速度比高斯-赛德尔（GS）方法快得多。结论：与GS迭代方法相比，基于离散化方法的GMRES效率更高。

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