On the numerical treatment of hyperbolic differential equations with constant coefficients, particularly the n-dimensional wave equation

1969 ◽  
pp. 154-159 ◽  
Author(s):  
Karl Graf Finck von Finckenstein
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Numerical Treatment ◽  
Constant Coefficients ◽  
Hyperbolic Differential Equations ◽  
Dimensional Wave

Cauchy problems for second order hyperbolic differential equations with constant coefficients

1983 ◽  
Vol 26 (3) ◽  
pp. 307-311 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Second Order ◽  
Cauchy Problems ◽  
Order Linear ◽  
Constant Coefficients ◽  
Darboux Equation ◽  
One To One ◽  
Hyperbolic Differential Equations

1. It is well known [1] that there is a one-to-one relation between solutions of the Darboux equation and the wave equation. The purpose of this paper is to show that some recent results in the fractional calculus can be used to obtain a similar connection between solutions of Darboux's equation and second order linear hyperbolic differential equations with constant coefficients.

scholarly journals Some applications of the Riesz potential to the theory of the electromagnetic field and the meson field

1946 ◽  
Vol 188 (1012) ◽  
pp. 18-31 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Differential Equations ◽  
Riesz Potential ◽  
Analytical Continuation ◽  
Neutral Meson ◽  
Meson Field ◽  
Hyperbolic Differential Equations ◽  
Proper Energy

This paper contains some applications of the method of Marcel Riesz in the solution of normal hyperbolic differential equations, in particular the wave equation, where the known difficulties, due to the occurrence of divergent integrals, are avoided by a process of analytical continuation. In the theory of the electromagnetic field the method yields simple deductions of classical results, but also the results recently obtained by Dirac regarding the proper energy and proper momentum of an electron are obtained without any addition of new assumptions. The corresponding problem in Bhabha’s analogous theory for the neutral meson field are also studied.

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

2018 ◽  
Vol 73 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Wei Feng ◽  
Songlin Zhao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Symmetry Group ◽  
Group Method ◽  
Radial Solutions ◽  
Radial Wave ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Wave ◽  
Radial Wave Equation

AbstractIn this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

scholarly journals A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables

2019 ◽  
Vol 27 (1) ◽  
pp. 25-34
Author(s):  
Sora Otsuki ◽  
Pauline N. Kawamoto ◽  
Hiroshi Yamazaki
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Derivative ◽  
Separation Of Variables ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Partial Differential Equations ◽  
Dimensional Wave

Summary In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.

scholarly journals On some solutions of second order hyperbolic differential equations with constant coefficients

1987 ◽  
Vol 29 (1) ◽  
pp. 69-72
Author(s):  
J. S. Lowndes
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Second Order ◽  
Hyperbolic Differential Equation ◽  
Constant Coefficients ◽  
Image Position ◽  
Hyperbolic Differential Equations

If we seek solutions of the hyperbolic differential equationwhich depend only on the variables i and , we see that these solutions must be even in r and satisfy the differential equationThe object of this paper is to show that some recent results in the fractional calculus can be used to prove the following theorem.

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