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Taylor-Galerkin method for solving higher-order nonlinear complex differential equations.

Md Humayun Kabir1, Md Shafiqul Islam2, Md Kamrujjaman3

  • 1Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman University, Kishoreganj 2300, Bangladesh.

Methodsx
|January 6, 2025
PubMed
Summary
This summary is machine-generated.

This study introduces a Galerkin approach using Taylor polynomials to solve higher-order Complex Differential Equations (CDEs). The method offers a robust numerical solution with detailed error analysis for both linear and nonlinear CDEs.

Keywords:
35K57 (primary)35K6137N25Complex differential equationsFinite Element MethodGalerkin methodMSC: 92D25Residual error correctionTaylor polynomials

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Complex Analysis

Background:

  • Complex Differential Equations (CDEs) pose significant challenges in numerical computation.
  • Existing methods for solving CDEs have limitations in accuracy and applicability.
  • Higher-order CDEs require specialized numerical techniques for efficient resolution.

Purpose of the Study:

  • To present a novel Galerkin approach for numerically resolving higher-order Complex Differential Equations (CDEs).
  • To utilize Taylor polynomial functions as basis and weighted functions within the Galerkin framework.
  • To provide a comprehensive error analysis and comparative study of the proposed method.

Main Methods:

  • The Galerkin method is applied over a rectangular domain in the complex plane.
  • Taylor polynomial functions are employed as basis and weighting functions.
  • The Complex Differential Equation is transformed into a matrix equation, leading to systems of linear or nonlinear equations for Taylor coefficients.

Main Results:

  • The proposed Galerkin approach effectively converts Complex Differential Equations into matrix equations.
  • Numerical results demonstrate the method's accuracy when compared to existing Taylor and Bessel Collocation methods for linear CDEs.
  • Comparisons with exact solutions for nonlinear CDEs validate the proposed method's efficacy, with detailed error analysis presented graphically and in tabular form.

Conclusions:

  • The Galerkin approach with Taylor polynomials provides an effective numerical technique for solving higher-order Complex Differential Equations.
  • The method's matrix formulation and iterative techniques efficiently determine unknown Taylor coefficients.
  • The study confirms the proposed method's accuracy and reliability through rigorous error analysis and comparative studies.