We characterize the set of functions u0 ∈ L (R) such that the solution of the problem ut = Lu in R n × (0,∞) starting from u0 satisfy upper and lower bounds of the form c(1 + t) ≤ ‖u(t)‖2 ≤ c ′(1 + t) . Here L is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discus...

In this paper we have solved some temporal fractional functional equations in the sense of Caputo by a numerical method called SOME BLAISE ABBO(SBA). Unlike classical methods, bypasses discretization. Despite its youth, it has already proven itself. Indeed, accuracy and efficiency been solution ODEs PDEs with integer derivative. Its application to constitutes an important scientific contributio...

A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schrödinger type and on a relativistically-invariant system of PDEs of Klein–Gordon type. Isochronous variants of these evolution PDEs are also considered.

the complex-step derivative approximation is applied to compute numerical derivatives. in this study, we propose a new formula of fractional complex-step method utilizing jumarie definition. based on this method, we illustrated an approximate analytic solution for the fractional cauchy-euler equations. application in image denoising is imposed by introducing a new fractional mask depending on s...

We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein–Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. ...

A. We consider the semi-linear elliptic PDEs driven by the fractional Laplacian: { (−∆)su = f (x, u), in Ω, u = 0, in Rn\Ω. By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without AmbrosettiRabinowitz condition for non-local el...

We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDEs of KdV type, including generalized KdV and Benjamin–Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonloc...

Efficient Spectral-Galerkin algorithms are developed to solve multi-dimensional fractional elliptic equations with variable coefficients in conserved form as well as non-conserved form. These algorithms are extensions of the spectral-Galerkin algorithms for usual elliptic PDEs developed in [24]. More precisely, for separable FPDEs, we construct a direct method by using a matrix diagonalization ...

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of nonlinear PDEs linearizable by the matrix Hopf-Cole substitution (the Bürgers hierarchy). We derive examples of four-dimensional nonlinear matrix PDEs together wi...