Piecewise Analytic Solutions of Mixed Boundary Value Problems

1966 ◽  
Vol 18 ◽  
pp. 1121-1147 ◽  
Author(s):  
M. Eisen
Keyword(s):  
Differential Equations ◽  
Mixed Problem ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Analytic Solutions ◽  
General Criterion ◽  
Method Of Images ◽  
Correct Boundary Conditions ◽  
Hyperbolic Differential Equations ◽  
Boundary Surfaces

In a mixed problem one is required to find a solution of a system of partial differential equations when the values of certain combinations of the derivatives are given on two or more distinct intersecting surfaces. If the differential equations arise from some physical process, the correct boundary conditions are usually apparent. Many particular problems of this type have been solved by special methods such as separation of variables and the method of images. However, no general criterion has been given for what constitutes a correctly set mixed problem. In fact such problems have usually been formulated in connection with hyperbolic differential equations with data prescribed on two surfaces (called the initial and boundary surfaces).

scholarly journals Finite Difference and Iteration Methods for Fractional Hyperbolic Partial Differential Equations with the Neumann Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Fadime Dal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Mixed Problem ◽  
Variational Iteration Method ◽  
Analytic Solutions ◽  
Hyperbolic Partial Differential Equations ◽  
Neumann Condition ◽  
Stability Estimates ◽  
Partial Differential

The numerical and analytic solutions of the mixed problem for multidimensional fractional hyperbolic partial differential equations with the Neumann condition are presented. The stable difference scheme for the numerical solution of the mixed problem for the multidimensional fractional hyperbolic equation with the Neumann condition is presented. Stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional hyperbolic partial differential equations. He's variational iteration method is applied. The comparison of these methods is presented.

Oscillation Properties of Weakly Time Dependent Hyperbolic Equations

1983 ◽  
Vol 26 (3) ◽  
pp. 368-373
Author(s):  
Kurt Kreith
Keyword(s):  
Differential Equations ◽  
Time Dependence ◽  
Comparison Theorem ◽  
Hyperbolic Equations ◽  
Separation Of Variables ◽  
Time Dependent ◽  
Hyperbolic Differential Equations ◽  
Oscillation Properties

AbstractA Sturmian comparison theorem is established for a pair of linear hyperbolic differential equations. While the equations may be time dependent (in the sense of not allowing a separation of variables), a measure of the strength of such time dependence enters into the hypotheses of the theorem.

A Mixed Problem for Normal Hyperbolic Linear Partial Differential Equations of Second Order

1957 ◽  
Vol 9 ◽  
pp. 141-160 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Linear Partial Differential Equations ◽  
Great Generality ◽  
Modern Treatment ◽  
Hyperbolic Differential Equations

In the theory of hyperbolic differential equations a mixed boundary value problem involves two types of auxiliary conditions which may be described as initial and boundary conditions respectively. The problem of Cauchy, in which only initial conditions are present, has been studied in great detail, starting with the early work of Riemann and Volterra, and the well-known monograph of Hadamard (4). A modern treatment of great generality has been given by Leray (7).

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

scholarly journals Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Ihor Borachok ◽  
Roman Chapko ◽  
B. Tomas Johansson
Keyword(s):  
Cauchy Problem ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Single Layer ◽  
Normal Derivative ◽  
Boundary Integral ◽  
Iteration Step ◽  
3 Dimensional ◽  
Boundary Surfaces

AbstractWe consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [

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