Electrostatics in a gravitational field

1978 ◽  
Vol 80 (3-4) ◽  
pp. 201-211 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Gravitational Field ◽  
Closed Form ◽  
Electrostatic Potential ◽  
Point Charge ◽  
Elementary Solution ◽  
Elliptic Type ◽  
Partial Differential

SynopsisIn this paper, the electrostatic potential of a point charge in a Reisser-Nordström gravitational field is found in closed form by using the theory of Hadamard's elementary solution of a partial differential equation of elliptic type.

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

New Classical Solutions to F4 Theory

1981 ◽  
Vol 34 (2) ◽  
pp. 113 ◽  
Author(s):  
EA Jeffery
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Closed Form ◽  
Minkowski Space ◽  
Normal Curvature ◽  
Classical Solutions ◽  
Partial Differential ◽  
Space Problem

The partial differential equation 02rf>+m2c/>+Ac/>3 = 0 (where 0 2 = 0;-\72) is converted to the ordinary differential equation 2�>/du2 + (Nlu)d�>ldu +m2c/> +).�>3 = O. In this conversion, 02U and (OU)2 are assumed to depend only on u, where u is a Fnction of x,y, Z, t, and further, Nlu is the normal curvature of the hypersurface u = constant. In order to solve for u, the analogous Euclidean 4-space problem with 0 2 = 0; +\72 is examined initially. The only possible values of N are then 0, 1, 2, 3 corresponding to hypersurfaces that are a hyperplane, a hypercylinder (one curved face), a hypercylinder (two curved faces) and a ypersphererespectively. These hypersurfaces are transformed to Minkowski space via xk ~ ixk , k = 1,2,3. Then by solving the ordinary differential equation new solutions to the original partial differential equation are found, one of which has a closed form.

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.

scholarly journals On the behaviour near the boundary of solutions of a semi-linear partial differential equation of elliptic type

1981 ◽  
Vol 31 (4) ◽  
pp. 405-414
Author(s):  
J. Chabrowski ◽  
B. Thompson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Sobolev Space ◽  
Linear Partial Differential Equation ◽  
Elliptic Type ◽  
Partial Differential

AbsrtactThe purpose of this note is to investigate traces of the functions In(1 +|u|), where u is a solution of a semi-linear partial differential equation of elliptic type, belonging to an appropriate Sobolev space. This article complements the results of Chabrowski and Thompson (1980), and Mihailov (1972), (1976).

scholarly journals Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 24
Author(s):  
Muhammad Shakeel ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Closed Form ◽  
Fractional Order ◽  
Soliton Solutions ◽  
Wave Transformation ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Closed Form Solutions

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

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.

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