V.—On an Elementary Solution of a Partial Differential Equation of Parabolic Type. Part II: The Nature of the Singularity.

1941 ◽  
Vol 61 (1) ◽  
pp. 54-60 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Parabolic Type ◽  
Elementary Solution ◽  
Partial Differential ◽  
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SummaryIt is shown that the elementary solution Γ(x, ξ; t – τ) of the equationbehaves, as t → τ + O, in very much the same manner as the elementary solution of the equation of heat.

scholarly journals On the “elementary” solution of Laplace's Equation

1931 ◽  
Vol 2 (3) ◽  
pp. 135-139 ◽  
Author(s):  
H. S. Ruse
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Elementary Solution ◽  
General Linear ◽  
Laplace’S Equation ◽  
Partial Differential ◽  
Laplace's Equation ◽  
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Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

Monotone methods and attractivity results for Volterra integro-partial differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
Andrea Schiaffino ◽  
Alberto Tesei
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Past History ◽  
Main Tool ◽  
Parabolic Type ◽  
Supremum Norm ◽  
Partial Differential ◽  
Diffusion Effects ◽  
Monotone Methods ◽  
Globally Attractive

SynopsisA Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

The fundamental solution for the axially symmetric wave equation II

1980 ◽  
Vol 87 (3) ◽  
pp. 515-521
Author(s):  
Albert E. Heins
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Fundamental Solution ◽  
Free Space ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Symmetric Wave ◽  
Alternate Forms ◽  
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In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equationcould be written asif b > rorif r > b.

On a Linear Partial Differential Equation of Hyperbolic Type

1922 ◽  
Vol 41 ◽  
pp. 76-81
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Partial Differential Equation ◽  
Second Order ◽  
Hyperbolic Type ◽  
Order Partial Differential Equation ◽  
Sound Waves ◽  
Partial Differential ◽  
Method Of Solution ◽  
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Riemann's method of solution of a linear second order partial differential equation of hyperbolic type was introduced in his memoir on sound waves. It has been used by Darboux in discussing the equationwhere α, β, γ are functions of x and y.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
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On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

Existence Theorem for the Initial-Boundary Value Problem for a Singular Parabolic Partial Differential Equation

1972 ◽  
Vol 15 (2) ◽  
pp. 229-234
Author(s):  
Julius A. Krantzberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Image Position ◽  
Initial Boundary Value

We consider the initial-boundary value problem for the parabolic partial differential equation1.1in the bounded domain D, contained in the upper half of the xy-plane, where a part of the x-axis lies on the boundary B(see Fig.1).

On the integrals of a partial differential equation

1947 ◽  
Vol 43 (3) ◽  
pp. 348-359 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Partial Differential ◽  
One Dimensional ◽  
First Order ◽  
Progressive Waves ◽  
Image Position ◽  
Straight Lines ◽  
Dimensional Wave

The ordinary one-dimensional wave equationhas special integrals of the formwhich satisfy the first-order equationsrespectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight linesin an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

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