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On the Chaplyghin method for first order partial differential equations

Puźniakowska E.

Opuscula Mathematica

2008

Vol. 28, no. 2

163-178

Classical solutions of initial problems for nonlinear first order partial differential equations are considered. It is shown that under natural assumptions on given functions, there exist Chaplyghin sequences and they are convergent. Error estimates for approximate solutions are given. The method of characteristics is used for the construction of approximate solutions.

Difference methods for infinite systems of hyperbolic functional differential equations on the Haar pyramid

Jaruszewska-Walczak D.

Opuscula Mathematica

2004

Vol. 24, no. 1

85-96

We consider the Cauchy problem for infinite system of differential functional equations ∂tzk(t, x) = fk(t, x, z, ∂xzk(t, x)), k mem N. In the paper we consider a general class of difference methods for this problem. We prove the convergence of methods under the assumptions that given functions satisfy the nonlinear estimates of the Perron type with respect to functional variables. The proof is based on functional difference inequalities. We constructed the Euler method as an example of difference method.

Numerical approximation of first order partial differential equations with deviated variables

Baranowska A., Kamont Z.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2003

[Z] 43(1)

1-31

Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference equations. The proof of the stability of the difference problem is based on a comparison technique with nonlinear estimates of the Perron type. This new approach to the numerical solving of nonlinear equations with deviated variables is generated by a quasilinearization method for initial problems. Numerical examples are given.