A new framework for polynomial approximation to differential equations

Area of Science:

• Numerical Analysis
• Differential Equations
• Computational Mathematics

Background:

• Initial value problems for differential equations require robust numerical solutions.
• Existing approximation methods may have limitations in handling complex equation types.
• Polynomial approximation offers a flexible approach for differential equation solutions.

Purpose of the Study:

• To present a novel framework for polynomial approximation of differential equation solutions.
• To extend the framework to address constant delay differential equations.
• To demonstrate the derivation of Runge-Kutta methods within this framework.

Main Methods:

• Expansion of the vector field along an orthonormal basis.
• Utilization of perturbation results for approximation accuracy.
• Generalization of the framework for ordinary and delay differential equations.

Main Results:

• A generalized framework for polynomial approximation of differential equations.
• Successful extension to constant delay differential equations.
• Derivation of relevant Runge-Kutta methods from the framework.

Conclusions:

• The proposed framework provides an effective method for approximating solutions to differential equations.
• The generalization enhances applicability to problems with constant delays.
• This approach facilitates the development of new numerical integration methods.