In this paper, the author investigates the scaling limit of a partial difference equation on the d dimensional integer lattice Zd, corresponding to a translation invariant random walk perturbed by a random vector field. In the case when the translation invariant walk scales to a Cauchy process he proves convergence to an effective equation on Rd. The effective equation corresponds to a Cauchy p...

in this paper, using the fixed point and direct methods, we prove the generalized hyers-ulam-rassias stability of the following cauchy-jensen additive functional equation: begin{equation}label{main} fleft(frac{x+y+z}{2}right)+fleft(frac{x-y+z}{2}right)=f(x)+f(z)end{equation} in various normed spaces. the concept of hyers-ulam-rassias stability originated from th. m. rassias’ stability theorem t...

In this paper, we present Legendre wavelet method to obtain numerical solution of a singular integro-differential equation. The singularity is assumed to be of the Cauchy type. The numerical results obtained by the present method compare favorably with those obtained by various Galerkin methods earlier in the literature.

in this article, a numerical approximation to the solution of the newell-whitehead equation (nwe) and cauchy problem of ill-posed non-linear diffusion equation have been studied. the presented scheme is obtained by using the derivative of the cubic b-spline quasi-interpolation (bsqi) to approximate the spatial derivative of the dependent variable and first order forward difference to approximat...

The main purpose of this article is to present an approximation method of for singular integrodifferential equations with Cauchy kernel in the most general form under the mixed conditions in terms of the second kind Chebyshev polynomials. This method transforms mixed singular integro-differential equations with Cauchy kernel and the given conditions into matrix equation and using the zeroes of ...

To every Darboux integrable system there is an associated Lie group G which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the Vessiot group G. If the Vessiot group G is solvable then the Cauchy problem can...

In this paper, using the fixed point and direct methods, we prove the generalized Hyers-Ulam-Rassias stability of the following Cauchy-Jensen additive functional equation: begin{equation}label{main} fleft(frac{x+y+z}{2}right)+fleft(frac{x-y+z}{2}right)=f(x)+f(z)end{equation} in various normed spaces. The concept of Hyers-Ulam-Rassias stability originated from Th. M. Rassias’ stability theorem t...

1. Linearity and Continuity 1.1 Continuity 1.2 Linearity 1.3 Perturbation theory and linearity 1.4 Axiomatically linear equations 1.4.1 Fields, Maxwell equations 1.4.2 Densities on phase space in classical physics 1.4.3 Quantum mechanics and Schrödinger equation 2. Examples 2.1 Ordinary differential equations 2.2 The Laplace equation 2.3 The wave equation 2.4 The heat equation and Schrödinger e...