Humayun Kabir Md, Shafiqul Islam Md, Kamrujjaman Md
Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman University, Kishoreganj 2300, Bangladesh.
Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh.
MethodsX. 2024 Dec 12;13:103078. doi: 10.1016/j.mex.2024.103078. eCollection 2024 Dec.
The Galerkin approach for numerically resolving higher-order Complex Differential Equations (CDEs) in a rectangular domain in the complex plane is presented in this work. Taylor polynomial functions are used in this method as basis or weighted functions. The CDE is converted into a matrix equation by employing the proposed method. A system of linear and nonlinear equations with unknown Taylor coefficients for linear and nonlinear CDEs, respectively, is represented by the resultant matrix equation. Results pertaining to this method's error analysis are discussed. The existing Taylor and Bessel Collocation methods are compared with the numerical results of the proposed method for linear CDEs, and the existing exact solutions and numerical results of the proposed method for nonlinear CDEs are also compared. The comparative results are displayed graphically for the real ( ) and imaginary ( ) parts, respectively, as well as in tabular form containing absolute error and maximum absolute error . The methodology of this study focused on the Galerkin integral domain which is a rectangle shape in the complex plane and Taylor polynomial is the shape function. Matrix formulation procedure and iterative technique are implemented to find out the undetermined Taylor coefficients.
本文提出了一种在复平面矩形域中数值求解高阶复微分方程（CDEs）的伽辽金方法。该方法使用泰勒多项式函数作为基函数或加权函数。通过所提出的方法将CDE转化为矩阵方程。所得矩阵方程分别表示了线性和非线性CDEs的具有未知泰勒系数的线性和非线性方程组。讨论了该方法误差分析的相关结果。将现有的泰勒和贝塞尔配置方法与所提方法对线性CDEs的数值结果进行了比较，同时也比较了所提方法对非线性CDEs的现有精确解和数值结果。分别以图形方式展示了实部（ ）和虚部（ ）的比较结果，以及以表格形式列出了绝对误差 和最大绝对误差 。本研究的方法聚焦于复平面中为矩形形状的伽辽金积分域，且泰勒多项式为形函数。通过实施矩阵公式化过程和迭代技术来求解未确定的泰勒系数。
