<p>In this paper, we construct high accuracy numerical methods for solving third, fourth and fifth order nonlinear functional differential equations (FDE). The methods are based on the discretization of iterative procedures formulated at the continuous level, incorporating trapezoidal quadrature formulas with correction terms and high-order interpolation. Depending on the number of correction terms and the degree of interpolation, we obtain methods of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2193_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$O(h^4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2193_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$O(h^6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> accuracy for third and fourth order FDEs, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2193_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$O(h^6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> accuracy for fifth order FDEs. Several numerical experiments are presented to validate the theoretical results. The proposed approach is general and can be applied to FDEs of arbitrary order.</p>

High accuracy numerical methods for solving high order functional differential equations

期刊论文
发表日期：

2025年7月31日

查看全文

Dang Quang A,
Dang Quang Long,
Vu Vinh Quang

摘要

In this paper, we construct high accuracy numerical methods for solving third, fourth and fifth order nonlinear functional differential equations (FDE). The methods are based on the discretization of iterative procedures formulated at the continuous level, incorporating trapezoidal quadrature formulas with correction terms and high-order interpolation. Depending on the number of correction terms and the degree of interpolation, we obtain methods of $O(h^4)$ $O (h^{4})$ and $O(h^6)$ $O (h^{6})$ accuracy for third and fourth order FDEs, and $O(h^6)$ $O (h^{6})$ accuracy for fifth order FDEs. Several numerical experiments are presented to validate the theoretical results. The proposed approach is general and can be applied to FDEs of arbitrary order.