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High order numerical methods for solving high orders functional differential equations

Dang Quang A, Dang Quang Long

Abstract

In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of $O(h^4)$ and $O(h^6)$ accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.

High order numerical methods for solving high orders functional differential equations

Abstract

In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of and accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.

Paper Structure

This paper contains 12 sections, 3 theorems, 57 equations, 22 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iterative method on continuous level for FDE
  4. Trapezoidal quadrature formula with corrections
  5. Construction of numerical methods of orders 4 and 6 for the third order FDE
  6. Computing the integrals of the type in \ref{['iter2c']}
  7. Discrete iterative method for the BVP \ref{['eq3']}-\ref{['eq4']}
  8. Examples for third order FDEs
  9. Construction of fourth and sixth order accuracy for fourth order FDE
  10. Examples for fourth order FDEs
  11. On high order numerical methods for fifth order FDE
  12. Concluding remarks

Key Result

Theorem 2.1

(see Theorems 2.2 and 3.1 in Dang1) Assume that: Then: i) The the problem eq3-eq4 has a unique solution $u(t) \in C^3[0,a]$, satisfying the estimate ii) The above iterative method converges and there holds the estimate where $u$ is the exact solution of the problem and

Theorems & Definitions (3)

  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2