Related papers: On the Laplace equation in d-dimension
This is not a new result. Purpose of this work is to describe a method to search the analytical expression of the general real solution of the two-dimensional Laplace differential equation. This thing is not easy to find in scientific…
We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure…
For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion…
We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations…
We derive an expansion for the fundamental solution of Laplace's equation in flat-ring coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and…
In this paper, we solve Laplace equation analytically by using differential transform method. For this purpose, we consider four models with two Dirichlet and two Neumann boundary conditions and obtain the corresponding exact solutions. The…
It is well-known that any solution of the Laplace equation is a real or imaginary part of a complex holomorphic function. In this paper, in some sense, we extend this property into four order hyperbolic and elliptic type PDEs. To be more…
The general d-dimensional twisted group lattice is solved. The irreducible representations of the corresponding group are constructed by an explicit procedure. It is proven that they are complete. All matrix representation solutions to the…
Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf…
We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form…
It is proved the existence of nonclassical solutions of the Neumann and Poincare problems for generalizations of the Laplace equation in anisotropic and nonhomogeneous media in almost smooth domains with arbitrary boundary data that are…
A general solution to the Complex Bateman equation in a space of arbitrary dimensions is constructed.
We present the spline-interpolation approximate solution of the Dirichlet problem for the Laplace equation in the bodies of revolution, cones and cylinders. Our method is based on reduction of the 3D problem to the sequence of 2D Dirichlet…
We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental…
Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with…
The Laplace equation in three dimensional Euclidean space is $R$-separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lam\'e-Wangerin functions called internal and external bi-cyclide harmonics. An…
We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions…
In this article we give in analytical closed form the solutions of the Direchlet problems for the Laplace equations with inverse square and singular P\"oschl-Teller potentials
We derive eigenfunction expansions for a fundamental solution of Laplace's equation in three-dimensional Euclidean space in 5-cyclidic coordinates. There are three such expansions in terms of internal and external 5-cyclidic harmonics of…
The solution in hyperspherical coordinates for $N$ dimensions is given for a general class of partial differential equations of mathematical physics including the Laplace, wave, heat and Helmholtz, Schr\"{o}dinger, Klein-Gordon and…