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 On solving initial value problems for partial differential equations in maple

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Srinivasarao Thota
Keyword(s):  
Applied Sciences ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Partial Differential Operator ◽  
Initial Value ◽  
Partial Differential ◽  
Greens Function ◽  
The Given

Abstract Objectives In this paper, we present and employ symbolic Maple software algorithm for solving initial value problems (IVPs) of partial differential equations (PDEs). From the literature, the proposed algorithm exhibited a great significant in solving partial differential equation arises in applied sciences and engineering. Results The implementation include computing partial differential operator (), Greens function () and exact solution () of the given IVP. We also present syntax, , to apply the partial differential operator to verify the solution of the given IVP obtained from . Sample computations are presented to illustrate the maple implementation.

scholarly journals An uncoupling procedure for a class of coupled linear partial differential equations

1985 ◽  
Vol 26 (4) ◽  
pp. 503-516 ◽  
Author(s):  
A. McNabb
Keyword(s):  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Large Class ◽  
Partial Differential Operator ◽  
Coupled System ◽  
System Of Equations ◽  
Partial Differential ◽  
Similar System ◽  
Linear Partial Differential Equations

AbstractA Fredholm operator exists which maps the solutions of a system of linear partial differential equations of the form ∂u/∂t = DLu + Au coupled by a matrix A onto those solutions of a similar system coupled by a matrix B which have the same initial values. The kernels of this operator satisfy a hyperbolic system of equations. Since these equations are independent of the linear partial differential operator L, the same operator serves as a mapping for a large class of equations. If B is chosen diagonal, the solutions of a coupled system with matrix A may be obtained from the uncoupled system with matrix B.

Mixed Problems for Linear Systems of first Order Equations

1958 ◽  
Vol 10 ◽  
pp. 127-160 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Auxiliary Data ◽  
First Order ◽  
Independent Variables ◽  
Mixed Problems ◽  
Linear Partial Differential Equations ◽  
Data Problem ◽  
Hyperbolic Differential Equations

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

scholarly journals Correct solvability and representation of the solutions of Volterra integro-differential equations with exponential-fractional kernels

2019 ◽  
Vol 488 (5) ◽  
pp. 476-480
Author(s):  
V. V. Vlasov ◽  
N. A. Rautian
Keyword(s):  
Hilbert Space ◽  
Spectral Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Unbounded Operator ◽  
Strong Solutions ◽  
Abstract Form ◽  
Partial Differential ◽  
Operator Functions ◽  
Well Posed

For abstract integro-differential equations with unbounded operator coefficients in a Hilbert space, we study the well-posed solvability of initial problems and carry out spectral analysis of the operator functions that are symbols of these equations. This allows us to represent the strong solutions of these equations as series in exponentials corresponding to points of the spectrum of operator functions. The equations under study are the abstract form of linear integro-partial differential equations arising in viscoelasticity and several other important applications.

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