Classical Solution Of Mixed Problem Research Articles

Classical solution of the mixed problem for a nonlinear equation

The first mixed problem for a nonlinear equation is considered in the quarter plane. The Cauchy conditions are set at the bottom of the boundary. The Dirichlet condition is set on the left part of the boundary. The solution is constructed using the method of characteristics in an implicit analytical form as a solution of the integral equation. The solvability of these integral equations, the smoothness of the solutions, and their dependence on the initial data are investigated. The uniqueness is proved and the conditions are established, under which there exists a piecewise smooth and classical solution of the first mixed problem.

Well-Posedness of a Mixed Problem in a Cylindrical Domain for One Class of Multidimensional Hyperbolic-Parabolic Equations

We prove the unique solvability and obtain the explicit expression for the classical solution of mixed problem in a cylindrical domain for one class of multidimensional hyperbolic-parabolic equations.

Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions

We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional dierential equations. Dierence methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite dierence systems. Methods of second type are truncation of the infinite dierence system, so that the resulting dierence problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-dierenti al problems can be obtained from the general model by specifying the given operators.

Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain \(\Omega\) with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of \(\partial\Omega\) with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on \((0,c)\times\Omega\) to the initial boundary value problem, for small \(c\). Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

Implicit difference inequalities corresponding to parabolic functional differential equations

We give theorems on implicit difference inequalities generated by initial-boundary value problems for parabolic functional differential equations. We apply this result for the in- vestigation of the stability of difference schemes. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference methods. The proofs of the convergence of difference methods are based on a comparison technique and the results on difference functional inequalities are used. Numerical examples are presented.

Numerical method of bicharacteristics for hyperbolic partial functional differential equations

Classical solutions of mixed problems for first-order partial functional differential equations in several independent variables are approximated in this paper by solutions of a difference problem of Euler type. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. Convergence of explicit difference schemes is proved by consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type. Differential equations with deviated variables and differential integral equations can be obtained from the general model by specifying the given operators.

Implicit Difference Inequalities Corresponding to First‐Order Partial Differential Functional Equations

We give a theorem on implicit difference functional inequalities generated by mixed problems for nonlinear systems of first‐order partial differential functional equations. We apply this result in the investigations of the stability of difference methods. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference schemes. The proof of the convergence of difference method is based on comparison technique, and the result on difference functional inequalities is used. Numerical examples are presented.

Numerical method of characteristics for semilinear partial functional differential systems

Classical solutions of mixed problems for first order partial functional differential systems in several independent variables are approximated in the paper with solutions of a difference problem of the Euler type. The mesh for the approximate solutions is obtained by a numerical solving of equations of characteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from a general model by specializing given operators.