Software for the numerical solution of first-order partial differential equations
Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is d...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Peoples’ Friendship University of Russia (RUDN University)
2019-12-01
|
| Series: | Discrete and Continuous Models and Applied Computational Science |
| Subjects: | |
| Online Access: | http://journals.rudn.ru/miph/article/viewFile/22195/17415 |
| _version_ | 1811233183588417536 |
|---|---|
| author | Yaroslav Yu Kuziv |
| author_facet | Yaroslav Yu Kuziv |
| author_sort | Yaroslav Yu Kuziv |
| collection | DOAJ |
| description | Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system. |
| first_indexed | 2024-04-12T11:17:20Z |
| format | Article |
| id | doaj.art-eb0c9e0e8cfa4f2eb3104b30518865e6 |
| institution | Directory Open Access Journal |
| issn | 2658-4670 2658-7149 |
| language | English |
| last_indexed | 2024-04-12T11:17:20Z |
| publishDate | 2019-12-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Discrete and Continuous Models and Applied Computational Science |
| spelling | doaj.art-eb0c9e0e8cfa4f2eb3104b30518865e62022-12-22T03:35:28ZengPeoples’ Friendship University of Russia (RUDN University)Discrete and Continuous Models and Applied Computational Science2658-46702658-71492019-12-01271424810.22363/2658-4670-2019-27-1-42-4818087Software for the numerical solution of first-order partial differential equationsYaroslav Yu Kuziv0Peoples’ Friendship University of Russia (RUDN University)Partial differential equations of the first order, arising in applied problems of optics and optoelectronics, often contain coefficients that are not defined by a single analytical expression in the entire considered domain. For example, the eikonal equation contains the refractive index, which is described by various expressions depending on the optical properties of the media that fill the domain under consideration. This type of equations cannot be analysed by standard tools built into modern computer algebra systems, including Maple.The paper deals with the adaptation of the classical Cauchy method of integrating partial differential equations of the first order to the case when the coefficients of the equation are given by various analytical expressions in the subdomains G1, . . . , Gk , into which the considered domain is divided. In this case, it is assumed that these subdomains are specified by inequalities. This integration method is implemented as a Python program using the SymPy library. The characteristics are calculatednumerically using the Runge-Kutta method, but taking into account the change in the expressions for the coefficients of the equation when passing from one subdomain to another. The main functions of the program are described, including those that can be used to illustrate the Cauchy method. The verification was carried out by comparison with the results obtained in the Maple computer algebra system.http://journals.rudn.ru/miph/article/viewFile/22195/17415eikonalpartial differential equationsympy |
| spellingShingle | Yaroslav Yu Kuziv Software for the numerical solution of first-order partial differential equations Discrete and Continuous Models and Applied Computational Science eikonal partial differential equation sympy |
| title | Software for the numerical solution of first-order partial differential equations |
| title_full | Software for the numerical solution of first-order partial differential equations |
| title_fullStr | Software for the numerical solution of first-order partial differential equations |
| title_full_unstemmed | Software for the numerical solution of first-order partial differential equations |
| title_short | Software for the numerical solution of first-order partial differential equations |
| title_sort | software for the numerical solution of first order partial differential equations |
| topic | eikonal partial differential equation sympy |
| url | http://journals.rudn.ru/miph/article/viewFile/22195/17415 |
| work_keys_str_mv | AT yaroslavyukuziv softwareforthenumericalsolutionoffirstorderpartialdifferentialequations |