Solution Of Mixed Problem Research Articles

Mixed Problem with an Integral Two-Space-Variables Condition for a Class of Hyperbolic Equations

This paper is devoted to the proof of the existence and uniqueness of the classical solution of mixed problems which combine Neumann condition and integral two-space-variables condition for a class of hyperbolic equations. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered.

Exact solutions and numerical approximations of mixed problems for the wave equation with delay

This work deals with the construction of exact and continuous numerical solutions of mixed problems for the wave equation with delayutt(t,x)=a2uxx(t,x)+bu(t-τ,x),t>τ,0⩽x⩽l.Using the method of separation of variables, and obtaining explicit expressions for the solution of a general initial value problem for a second order delay differential equation, an exact infinite series solution is derived. Bounds on the truncation errors, when this series is approximated by finite sums, are given, thus providing constructive continuous numerical solutions with prescribed accuracy in bounded domains.

Defference schemes for weak solutions of mixed problem for a class of hyperbolic differential equations I.

Defference schemes for weak solutions of mixed problem for a class of hyperbolic differential equations I.

Difference schemes for weak solutions of mixed problem for a class of parabolic differential equations I.

Difference schemes for weak solutions of mixed problem for a class of parabolic differential equations I.

Examples of the nonuniqueness of solutions of the mixed problem for the heat equation in unbounded domains

For a wide class of domains of revolution, we construct examples of the nonuniqueness of solutions of the first mixed problem for the heat equation, which supports the exactness of a uniqueness class of Tacklind type.

Numerical solution of mixed problem of parabolic equation with an integral conditions by using finite difference and orthogonal function approximation

In this work the combined finite difference and orthogonal function approximation methods have been proposed for the numerical solution of the one-dimensional parabolic equation with two integrals conditions. The time variable is approximated using a finite difference scheme. But the orthogonal function approximation is employed for discretizing the space variable by using orthogonal cosine function.

Positive solutions for mixed problems of singular fractional differential equations

AbstractWe investigate the existence of positive solutions to the singular fractional boundary value problem: \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$\end{document}, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a Lq‐Carathéodory function, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$q > \frac{1}{\alpha -1}$\end{document}, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$^c \hspace{-1.0pt}D$\end{document} stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones.

Random mixed hyperbolic models: Numerical analysis and computing

AbstractThis paper deals with the construction of reliable numerical solutions of mixed problems for hyperbolic second order partial differential models with random information in the variable coefficients of the partial differential equation and in the initial data. Using random difference schemes a random discrete eigenfunctions method is developed in order to construct a discrete approximating stochastic process. Mean square consistency of the random difference scheme is treated and mean square stability of the numerical solution is studied and illustrated with examples. Statistical moments of the numerical solution are also computed.

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.

Unboundedness of solutions of one third-order hyperbolic equation

We have established conditions under which the solution of mixed problem for one third-order nonlinear hyperbolic equation becomes unbounded at a finite moment of time.

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.

Структура решения смешанной задачи для волнового уравнения на компактном геометрическом графе в случае ненулевой начальной скорости

Структура решения смешанной задачи для волнового уравнения на компактном геометрическом графе в случае ненулевой начальной скорости

Analytic-numerical solutions of diffusion mathematical models with delays

This work deals with the construction of analytic-numerical solutions of mixed problems for diffusion and reaction-diffusion equations with delay. Using the method of separation of variables, exact theoretical infinite series solutions are derived. Bounds on the truncation errors, when these series are approximated by finite sums, are given, thus providing constructive continuous numerical solutions with prescribed accuracy in bounded domains. In order to improve the computational efficiency of these numerical solutions, polynomial approximations to the initial functions are also considered.

A priori estimates and continuous dependence of solutions of mixed problems for parabolic equations as nonlocal boundary conditions pass into local ones

In the rectangle G = (0, 1) × (0, T), we consider the family of problems $$ \begin{gathered} \frac{1} {{a(x,t)}}\frac{{\partial u_\alpha }} {{\partial t}} - \frac{{\partial ^2 u_\alpha }} {{\partial x^2 }} = f(x,t), u_\alpha (x,0) = \phi _\alpha (x), u_\alpha (0,t) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ u_0 (1,t) = h(t), \frac{{\partial u_1 (1,t)}} {{\partial x}} = h(t), \frac{{u_\alpha (1,t) - u_\alpha (\alpha ,t)}} {{1 - \alpha }} = h(t), 0 < \alpha < 1, \hfill \\ a_1 \geqslant a(x,t) \geqslant a_0 > 0, h \in W_2^1 (0,T), \phi _\alpha \in W_2^1 (0,T), \phi _\alpha (0) = 0, 0 \leqslant \alpha \leqslant 1, \hfill \\ \phi _0 (1) = h(0), \phi '_1 (1) = h(0), \frac{{\phi _\alpha (1) - \phi _\alpha (0)}} {{1 - \alpha }} = h(0), 0 < \alpha < 1, f \in L_2 (G) \hfill \\ \end{gathered} $$ . It is well known that, for α = 0 and α = 1, the corresponding problems with local conditions are solvable, and the solutions are unique and belong to W 2 2,1 (G). We prove the existence and uniqueness of solutions of the family of problems with nonlocal conditions for each α ∈ (0, 1). For the differences u α − u 0 and u α − u 1 (0 < α < 1), we establish a priori estimates and use them to prove that if ϕ α → ϕ 0 as α → 0, then u α → u 0 and if ϕ α → ϕ 1 as α → 1, then u α → u 1.

Hidden regularity for the Kirchhoff equation

In this work we study the hidden regularity for solutions of mixed problem associated to the Kirchhoff model for smalldeformations of a membrane$u''-M(||u||^2) \Delta u=0,$when the initial data are considered in spaces with few regularity.

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.

A numerical method of bicharacteristics For quasi-linear partial functional Differential equations

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

Global solutions and smoothing effects for semi-linear evolution equations in circular domains

The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the corresponding region. A forced Cahn–Hilliard-type equation in a unit disc Ω is considered as an example. The current work focuses on revealing the mechanism of nonlinear smoothing, i.e., on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem. To this end convolutions of Rayleigh functions with respect to the Bessel index are employed. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424; V. Varlamov, Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains, J. Math. Anal. Appl. 327 (2007) 1461–1478]. In order to reveal the effect of nonlinear smoothing, anisotropic Sobolev spaces H s , α ( Ω ) are introduced. They are based on the Sobolev spaces H s ( Ω ) “weighted” by tangential derivatives, so that the index α is responsible for the smoothness in θ and s is the usual Sobolev index. Global-in-time solutions of the mixed problem in question are constructed, and additional smoothness with respect to the angular coordinate is established in the anisotropic Sobolev spaces. The above mentioned special functions are essentially used for this purpose.

On the absence of global solutions of nonlinear mixed problems

The paper gives examples demonstrating the blow-up phenomenon for solutions of partial differential equations. It is shown that the nature of blow-up of solutions of the mixed problem substantially differs from that of solutions of the Cauchy problem.