In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our method. Finally, three examples have been employed to illustrate the preciseness and effectiveness of the proposed method.

We establish a 3G-Theorem for the Green's function for an unbounded regular domain D in R n (n ≥ 3), with compact boundary. We exploit this result to introduce a new class of potentials K(D) that properly contains the classical Kato class K ∞ n (D). Next, we study the existence and the uniqueness of a positive continuous solution u inD of the following nonlinear singular elliptic problem      ∆u + ϕ(., u) = 0 , in D(in the sense of distributions)where ϕ is a nonnegative Borel measurable function in D × (0, ∞), that belongs to a convex cone which contains, in particular, all functions ϕ(x, t) = q(x)t −σ , σ ≥ 0 with q ∈ K(D). We give also some estimates on the solution u.

We establish a new form of the 3G theorem for polyharmonic Green function on the unit ball of R n (n ≥ 2) corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functions K m,n containing properly the classical Kato class K n . We exploit properties of functions belonging to K m,n to prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order 2m.

No abstract

In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to obtain the exact solution for nonlinear fractional problems, then we modified the double Laplace transform and combined it with the Adomian decomposition method. Later, we applied the new method to solve regular and singular conformable fractional coupled Burgers’ equations. Further, in order to illustrate the effectiveness of present method, we provide some examples.