Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays
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...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2020/2854574 |
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| author | Josef Rebenda Zuzana Pátíková |
| author_facet | Josef Rebenda Zuzana Pátíková |
| author_sort | Josef Rebenda |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b2932d3d34b64121b1aff6e66bc19f50 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-b2932d3d34b64121b1aff6e66bc19f502025-08-20T03:39:21ZengWileyComplexity1076-27871099-05262020-01-01202010.1155/2020/28545742854574Differential Transform Algorithm for Functional Differential Equations with Time-Dependent DelaysJosef Rebenda0Zuzana Pátíková1CEITEC BUT, Brno University of Technology, Purkyňova 656/123, 61200 Brno, Czech RepublicDepartment of Mathematics, Faculty of Applied Informatics, Tomas Bata University in Zlín, Nad Stráněmi 4511, Zlín 76005, Czech RepublicAn 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.http://dx.doi.org/10.1155/2020/2854574 |
| spellingShingle | Josef Rebenda Zuzana Pátíková Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays Complexity |
| title | Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays |
| title_full | Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays |
| title_fullStr | Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays |
| title_full_unstemmed | Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays |
| title_short | Differential Transform Algorithm for Functional Differential Equations with Time-Dependent Delays |
| title_sort | differential transform algorithm for functional differential equations with time dependent delays |
| url | http://dx.doi.org/10.1155/2020/2854574 |
| work_keys_str_mv | AT josefrebenda differentialtransformalgorithmforfunctionaldifferentialequationswithtimedependentdelays AT zuzanapatikova differentialtransformalgorithmforfunctionaldifferentialequationswithtimedependentdelays |