scholarly journals Teaching Fourier series, partial differential equations and their applications with help of computer algebra system

2009 ◽  
Vol 7 (1) ◽  
pp. 51-68
Author(s):  
Ildikó Perjési-Hámori ◽  
Csaba Sárvári
Keyword(s):  
Fourier Series ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Partial Differential

SFOPDES: A Stepwise First Order Partial Differential Equations Solver with a Computer Algebra System

2019 ◽  
Vol 78 (9) ◽  
pp. 3152-3164 ◽  
Author(s):  
José Luis Galán-García ◽  
Gabriel Aguilera-Venegas ◽  
Pedro Rodríguez-Cielos ◽  
Yolanda Padilla-Domínguez ◽  
María Ángeles Galán-García
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Partial Differential ◽  
First Order

scholarly journals Software for the numerical solution of first-order partial differential equations

2019 ◽  
Vol 27 (1) ◽  
pp. 42-48
Author(s):  
Yaroslav Yu Kuziv
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Computer Algebra ◽  
Partial Differential ◽  
First Order ◽  
Modern Computer ◽  
Runge Kutta Method ◽  
The Media ◽  
Cauchy Method ◽  
Analytical Expressions

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.

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