Solving linear and nonlinear problems using Taylor Series Method

Abstract

The article deals with the solution of technical initial value problems. To solve such problems, an analytical or numerical approach is possible. The analytical approach can provide an accurate result; however, it is not available for all problems and it is not entirely suitable for calculation on a computer, due to the limited numerical accuracy. For this reason, the numerical approach is preferred. This approach uses ordinary differential equations to approximate the continuous behaviour of the real-world system. There are many known numerical methods for solving such equations, most of them limited in their accuracy, have a limited region of stability and can be quite slow to achieve the acceptable solution. The numerical method proposed in this article is based on the Taylor series and overcomes the biggest challenge, i.e. calculating higher derivatives. The aim of the article is therefore twofold: to introduce the method and show its properties, and to show its behaviour when solving problems composed of linear and nonlinear ordinary differential equations. Linear problems are modelled by partial differential equations and solved in parallel using the PETSc library. The parallel solution is demonstrated using the wave equation, which is transformed into the system of ordinary differential equations using the method of lines. The solution of nonlinear problems is introduced together with several optimisations that significantly increase the calculation speed. The improvements are demonstrated using several numerical examples that are solved using MATLAB software.

The article deals with the solution of technical initial value problems. To solve such problems, an analytical or numerical approach is possible. The analytical approach can provide an accurate result; however, it is not available for all problems and it is not entirely suitable for calculation on a computer, due to the limited numerical accuracy. For this reason, the numerical approach is preferred. This approach uses ordinary differential equations to approximate the continuous behaviour of the real-world system. There are many known numerical methods for solving such equations, most of them limited in their accuracy, have a limited region of stability and can be quite slow to achieve the acceptable solution. The numerical method proposed in this article is based on the Taylor series and overcomes the biggest challenge, i.e. calculating higher derivatives. The aim of the article is therefore twofold: to introduce the method and show its properties, and to show its behaviour when solving problems composed of linear and nonlinear ordinary differential equations. Linear problems are modelled by partial differential equations and solved in parallel using the PETSc library. The parallel solution is demonstrated using the wave equation, which is transformed into the system of ordinary differential equations using the method of lines. The solution of nonlinear problems is introduced together with several optimisations that significantly increase the calculation speed. The improvements are demonstrated using several numerical examples that are solved using MATLAB software.