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.

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 The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions

1956 ◽  
Vol 8 ◽  
pp. 426-431 ◽  
Author(s):  
E. P. Miles ◽  
Ernest Williams
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Cauchy Data ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Image Position

We shall discuss solutions of linear partial differential equations of the form1where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;2j = 1,2, ߪ,s − 1,whenever the function P is annihilated by a finite iteration of the operator Φ.

Construction of metric and vector potential perturbations of a Reissner–Nordström black hole

1979 ◽  
Vol 369 (1736) ◽  
pp. 67-81 ◽  
Keyword(s):  
Black Hole ◽  
Partial Differential Equations ◽  
Linear System ◽  
Differential Equations ◽  
Vector Potential ◽  
Maxwell Equations ◽  
Coupled System ◽  
System Of Equations ◽  
Partial Differential ◽  
Explicit Formulae

When an appropriate decoupling of variables in a coupled linear system of partial differential equations is obtained, a recently described procedure enables one to construct solutions to the full coupled system of equations. We employ this procedure here to generate solutions of the linearized Einstein–Maxwell equations describing perturbations of a Reissner–Nordström black hole, using Chandrasekhar’s recent decoupling of these equations. Explicit formulae are given for the metric and vector potential perturbations for each parity type.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

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