Differential Equations of Time Dependent Order

A. Ludu

doi:10.1063/1.4964959

Differential Equations of Time Dependent Order

A. Ludu

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

We introduce a special type of ordinary differential equations d superscript x x / dt superscript x = f(t, x) whose order of differentiation is a continuous variable depending on the dependent x or independent t variables. We show that such variable order of differentiation equations (VODE) can be solved as Volterra integral equations of second kind with singular integrable kernel. We find the conditions for existence and uniqueness of solutions of such VODE, and present some numeric solutions for particular cases exhibiting bifurcations and blow-up.

Original language	American English
Title of host publication	Application of Mathematics in Technical and Natural Sciences: 8th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS'16
DOIs	https://doi.org/10.1063/1.4964959
State	Published - Oct 2016

Keywords

ordinary differential equations

Disciplines

Ordinary Differential Equations and Applied Dynamics

Access to Document

10.1063/1.4964959License: Unspecified

Cite this

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title = "Differential Equations of Time Dependent Order",

abstract = " We introduce a special type of ordinary differential equations d superscript x x / dt superscript x = f(t, x) whose order of differentiation is a continuous variable depending on the dependent x or independent t variables. We show that such variable order of differentiation equations (VODE) can be solved as Volterra integral equations of second kind with singular integrable kernel. We find the conditions for existence and uniqueness of solutions of such VODE, and present some numeric solutions for particular cases exhibiting bifurcations and blow-up.",

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N2 - We introduce a special type of ordinary differential equations d superscript x x / dt superscript x = f(t, x) whose order of differentiation is a continuous variable depending on the dependent x or independent t variables. We show that such variable order of differentiation equations (VODE) can be solved as Volterra integral equations of second kind with singular integrable kernel. We find the conditions for existence and uniqueness of solutions of such VODE, and present some numeric solutions for particular cases exhibiting bifurcations and blow-up.

AB - We introduce a special type of ordinary differential equations d superscript x x / dt superscript x = f(t, x) whose order of differentiation is a continuous variable depending on the dependent x or independent t variables. We show that such variable order of differentiation equations (VODE) can be solved as Volterra integral equations of second kind with singular integrable kernel. We find the conditions for existence and uniqueness of solutions of such VODE, and present some numeric solutions for particular cases exhibiting bifurcations and blow-up.

KW - ordinary differential equations

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DO - 10.1063/1.4964959

M3 - Chapter

BT - Application of Mathematics in Technical and Natural Sciences: 8th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS'16

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