Rainbow solutions to the Sidon equation

Rainbow solutions to the Sidon equation

We prove that every 4-coloring of [n] with each color class having cardinality more than n+1 6 contains a rainbow solution to x + y = z + w, and that this bound is tight.

On Rainbow Solutions to an Equation with a Quadratic Term

In this paper, we prove that every 3-coloring of the positive integers such that the upper density of each color is greater than 4 contains a rainbow solution to a−b = c 2. A solution is rainbow if all of its elements are of different colors. Furthermore, the 4 bound is sharp. We also prove two results for rainbow solutions of a− b = c2 in Zn. One stipulates that if Zn, for an odd n, is partiti...

Decay estimates of solutions to the IBq equation

‎In this paper we focus on the Cauchy problem for the generalized‎ ‎IBq equation with damped term in $n$-dimensional space‎. ‎We establish the global existence and decay estimates of solution with $L^q(1leq qleq 2)$ initial value‎, ‎provided that the initial value is suitably small‎. ‎Moreover‎, ‎we also show that the solution is asymptotic to the solution $u_L$ to the corresponding linear equa...

Analytical solutions for the fractional Fisher's equation

In this paper, we consider the inhomogeneous time-fractional nonlinear Fisher equation with three known boundary conditions. We first apply a modified Homotopy perturbation method for translating the proposed problem to a set of linear problems. Then we use the separation variables  method to solve obtained problems. In examples, we illustrate that by right choice of source term in the modified...

WAVELET SOLUTIONS OF THE KLEIN-GORDON EQUATION