A differential transformation approach for solving functional differential equations with multiple delays

Rebenda, Josef; Šmarda, Zdeněk

doi:10.1016/j.cnsns.2016.12.027

Cited by 20 publications

(13 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An introduction of a generalization of the differential transformation (DT) called the fractional differential transformation (FDT) is given in this subsection. For more details on DT and FDT, we recommend solid papers [7][8][9][10][11][12]. Definition 2.…”

Section: Fractional Differential Transformationmentioning

confidence: 99%

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Rebenda

2019

Symmetry

Self Cite

View full text Add to dashboard Cite

The differential transformation, an approach based on Taylor’s theorem, is proposed as convenient for finding an exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem gives a reliable and expected outcome including the phenomenon of symmetry in choice of orders of the differential transformation of the multi-order system.

show abstract

Section: Fractional Differential Transformationmentioning

confidence: 99%

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Rebenda

2019

Symmetry

Self Cite

View full text Add to dashboard Cite

show abstract

“…DT algorithm for the case t * � t 0 � 0 with all delays being proportional is described in [14]. DT algorithm for the case t * < t 0 when all delays are constant is introduced in [15]. In this paper, we develop the algorithm for the case t * < t 0 when at least one delay is nonconstant.…”

Section: Problem Statementmentioning

confidence: 99%

Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Rebenda

Pátíková

2020

Complexity

Self Cite

View full text Add to dashboard Cite

An algorithm using the differential transformation which is convenient for finding numerical solutions to initial value problems for functional differential equations is proposed in this paper. We focus on retarded equations with delays which in general are functions of the independent variable. The delayed differential equation is turned into an ordinary differential equation using the method of steps. The ordinary differential equation is transformed into a recurrence relation in one variable using the differential transformation. Approximate solution has the form of a Taylor polynomial whose coefficients are determined by solving the recurrence relation. Practical implementation of the presented algorithm is demonstrated in an example of the initial value problem for a differential equation with nonlinear nonconstant delay. A two-dimensional neutral system of higher complexity with constant, nonconstant, and proportional delays has been chosen to show numerical performance of the algorithm. Results are compared against Matlab function DDENSD.

show abstract

“…An introduction of a generalisation of the differential transformation (DT) called the fractional differential transformation (FDT) is given in this subsection. For more details on DT and FDT, we recommend sound papers [5], [6], [7], [8], [9], [10].…”

Section: Fractional Differential Transformationmentioning

confidence: 99%

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Rebenda¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

The differential transformation, an approach based on Taylor's theorem, is proposed as convenient for finding exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem results in a reliable and expected outcome.

show abstract

A differential transformation approach for solving functional differential equations with multiple delays

Cited by 20 publications

References 19 publications

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays

Application of Differential Transform to Multi-Term Fractional Differential Equations with Non-Commensurate Orders

Contact Info

Product

Resources

About