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

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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℜ</mml:mi> <mml:mi>e</mml:mi></mml:mrow> </mml:math> ) and imaginary ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ℑ</mml:mi> <mml:mi>m</mml:mi></mml:mrow> </mml:math> ) parts, respectively, as well as in tabular form containing absolute error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:math> and maximum absolute error <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mi>∞</mml:mi></mml:msup> <mml:mspace></mml:mspace> <mml:mi>n</mml:mi> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi></mml:mrow> </mml:math> . 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.