Abstract
We consider initial boundary-value problems of Dirichlet type for nonlinear equations. We give sufficient conditions for the convergence of a general class of one-step difference methods. We assume that the right-hand side of the equation satisfies an estimate of Perron type with respect to the functional argument.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Lakshmikantham and S. Leela,Differential and Integral Inequalities, New York, London (1969).
S. Bal, “Convergence of a difference method for a system of first-order partial differential equations of hyperbolic type,”Ann. Pol. Math.,30, 19–36 (1974).
Ho-Ling Chang, “Error estimations for certain approximate solutions of a nonlinear partial differential equation of the first order,”Ann. Pol. Math.,18, 293–298 (1966).
Z. Kamont and K. Przadka, “Difference methods for nonlinear partial differential equations of the first order,”Ann. Pol. Math.,48, 227–246 (1988).
Z. Kowalski, “A difference method for the nonlinear partial differential equation of the first order,”Ann. Pol. Math.,18, 235–242 (1966).
Z. Kowalski, “A difference method for certain hyperbolic systems of nonlinear partial differential equations of the first order,”Ann. Pol. Math.,19, 313–322 (1976).
G. Strang, “Accurate partial difference methods. II. Nonlinear problems,”Numer. Math.,6, 37–46 (1964).
P. Brandi, Z. Kamont, and A. Salvadori, “Approximate solutions of mixed problems for first-order partial differential-functional equations,”Atti Semin. Mat. Fis. Univ. Modena,39, 277–302 (1991).
K. Przadka, “Difference methods for nonlinear partial differential-functional equations of the first order,”Math. Nachr.,138, 105–123 (1988).
P. Brandi, Z. Kamont, and A. Salvadori, “Differential and difference inequalities related to mixed problems for first-order partial differential-functional equations,”Atti Semin. Mat. Fis. Univ. Modena,39, 255–276 (1991).
J. Szarski,Differential Inequalities, Warsaw (1967).
P. Brandi and R. Ceppitelli, “Approximate solutions (in a classic sense) for the Cauchy problem for semilinear hyperbolic systems,”Atti Semin. Mat. Fis. Univ. Modena,33, 77–112 (1984).
P. Brandi and R. Ceppitelli, “ɛ-Approximate solutions for hereditary nonlinear partial differential equations,”Rend. Circ. Mat. Palermo,2, No. 8, 399–417 (1981).
Z. Kamont and K. Przadka, “Difference methods for first-order partial differential-functional equations with initial-boundary conditions,”Zh Vych. Mat. Mat. Fiz.,31, 1476–1488 (1991).
K. M. Magomedov and A. S. Kholodov,Mesh-Characteristic Numerical Methods [in Russian], Nauka, Moscow (1988).
T. Meis and U. Marcowitz,Numerical Solution of Partial Differential Equations, New York (1961).
A. R. Mitchel and D. F. Griffiths,The Finite Difference Methods in Partial Differential Equations, New York, Toronto (1980).
H. I. Reinhardt,Analysis of Approximation Methods for Differential and Integral Equations, New York, (1985).
Author information
Authors and Affiliations
Additional information
Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 985–996, August, 1994.
Rights and permissions
About this article
Cite this article
Kamont, Z. Finite-difference approximation of first-order partial differential-functional equations. Ukr Math J 46, 1079–1092 (1994). https://doi.org/10.1007/BF01056169
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01056169