Optimal Solution of Linear Ordinary Differential Equations by Conjugate Gradient Method
For researchers and practitioners solving linear ODEs, this method offers a way to obtain more accurate solutions, but it is an incremental improvement over existing numerical methods.
The paper proposes a method to solve linear ODEs by representing the solution with Hermite cubic splines and minimizing the L2 norm of the residual using the conjugate gradient method, achieving smaller global error without additional mesh points.
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.