Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method

TL;DR

This paper introduces a novel approach to solving linear ODEs by formulating the problem as an optimization task minimized via the conjugate gradient method, leveraging spline interpolation and residual error minimization.

Contribution

It presents a new method combining spline interpolation and conjugate gradient optimization to find minimal residual solutions for linear ODEs, improving accuracy without extra mesh points.

Findings

Achieves smaller global error compared to traditional methods.

Utilizes sparsity of matrices for computational efficiency.

Effective for both initial and boundary value problems.

Abstract

Solving initial value problems and boundary value problems of Linear Ordinary Differential Equations (ODEs) plays an important role in many applications. There are various numerical methods and solvers to obtain approximate solutions represented by points. However, few work about optimal solution to minimize the residual can be found in the literatures. In this paper, we first use Hermit cubic spline interpolation at mesh points to represent the solution, then we define the residual error as the square of the L2 norm of the residual obtained by substituting the interpolation solution back to ODEs. Thus, solving ODEs is reduced to an optimization problem in curtain solution space which can be solved by conjugate gradient method with taking advantages of sparsity of the corresponding matrix. The examples of IVP and BVP in the paper show that this method can find a solution with smaller global error without additional mesh points.