An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space Wm 2 (G) are studied. The Fredholm property of the unbounded operator corresponding to the elliptic equation, acting on L2(G), and defined for functions from the space Wm 2 (G) that satis...

Consider a discrete uniformly elliptic divergence form equation on the d dimensional lattice Zd with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green’s function together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It ...

Elliptic problems appear naturally in physics mainly in two situations: as equations which describe equilibrium (for example, stationary solutions in General Relativity) and as constraints for the evolutions equations (for example, constraint equations in Electromagnetism and General Relativity). In addition, in General Relativity they appear often as gauge conditions for the evolutions equatio...

In this work, we employ the simple direct method to investigate the generalized mKdV equation with variable coefficients. By using this scheme, we found some new exact solutions of the equation, including four new types of Jacobi elliptic function solutions, and these solutions are degenerated to solitary wave solutions and triangle function solutions in the limit case when the modulus of the J...

In this paper, we consider an elliptic partial differential equation where a small parameter is multiplied with one or both of the second derivatives. Four types of basic spectral regularization methods such as Showalter’s, Tikhonov’s, Lardy’s and Lavrentiev’s are applied to approximate the solution by introducing another large (or small) parameter. Convergence of the regularized solutions to t...

(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper ...

For any elliptic curve E over a number field, there is, for each n ≥ 1, a symmetric n-power L-function, defined by an Euler product, and conjecturally having a meromorphic continuation and satisfying a precise functional equation. The sign in the functional equation is conjecturally a product of local signs. Given an elliptic curve over a finite extension of some Qp, we calculate the associated...

By using solutions of an ordinary differential equation, an auxiliary equationmethod is described to seek exact solutions of variablecoefficient KdV-MKdV equation. As a result, more new exact nontravelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, and combined Jacobi elliptic function solutions, for...

In this paper, we propose an iterative method based on the equation decomposition technique[11] for the numerical solution of a singular perturbation problem of fourth-order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second-order elliptic equation and a second-order singular perturbation problem. We prove that our approximate solution ...

A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn u,k . If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differenti...