scholarly journals Uniqueness of the solution of nonlinear totally characteristic partial differential equations

2005 ◽  
Vol 57 (4) ◽  
pp. 1045-1065
Author(s):  
Hidetoshi TAHARA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Uniqueness Of The Solution

scholarly journals On some stochastic parabolic differential equations in a Hilbert space

2005 ◽  
Vol 2005 (2) ◽  
pp. 167-173 ◽  
Author(s):  
Khairia El-Said El-Nadi
Keyword(s):  
Hilbert Space ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Partial Differential Operator ◽  
Parabolic Differential Equations ◽  
Partial Differential ◽  
Mixed Problems ◽  
Uniqueness Of The Solution ◽  
Weiner Process

We consider some stochastic difference partial differential equations of the form du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

scholarly journals CONVERGENCE SPEED OPTIMIZATION IN METHODS OF APPROXIMATE SOLVING OF INITIAL VALUES PROBLEM FOR THE SYSTEM OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

2018 ◽  
pp. 47-51
Author(s):  
A. I. Kazmerchuk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generalized Solution ◽  
Convergence Speed ◽  
Partial Differential ◽  
Approximate Methods ◽  
First Order ◽  
Initial Values ◽  
Quasilinear Partial Differential Equations ◽  
Uniqueness Of The Solution

In the theory of systems of quasilinear partial differential equations of the first order, the main questions are the solvability of initial values problem and justification of the approximate methods. This is due to problems in gas dynamics and hydromechanics. In the second half of the previous century attempts were made to construct a correct theory of solvability of problems or the systems of quasilinear partial differential equations of the first order. The necessity of the correct way of introductions the nothions of a generalized solution of initial values problems is connected with this. In this paper a class of systems of quasilinear partial differential equations of the first order is singled out for which the concept of a generalized solution is introduced. A method for constructing approximate methods for solving initial values problem is proposed. We obtained estimates of the convergence speed in approximate methods and proved the existence and uniqueness of the solution of initial values problem for systems of quasilinear partial differential equations of the first order of a certain form.

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