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 ◽  
Image Position

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.

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 ◽  
Image Position

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.

Cauchy's Problem in a Non-Linear Partial Differential Equation of Hyperbolic Type

1935 ◽  
Vol 31 (2) ◽  
pp. 195-202 ◽  
Author(s):  
M. Raziuddin Siddiqi
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Linear Partial Differential Equation ◽  
Positive Function ◽  
Hyperbolic Type ◽  
Partial Differential ◽  
Non Linear ◽  
Image Position

Let p (x) be an essentially positive function defined in the interval 0 ≤ x ≤ π. We consider the non-linear partial differential equationfor the boundary conditions u (x, t) = 0,for x = 0 and x = π,

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 ◽  
Image Position

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.

On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

1949 ◽  
Vol 1 (2) ◽  
pp. 191-198 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Arbitrary Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Partial Differential ◽  
Second Order Differential Equation ◽  
Image Position

The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

Sign in / Sign up

Export Citation Format

Share Document