Gas Dynamics type Burgers equation with convolutional nonlinearity

Abstract

In this paper, we consider an initial value problem for the Burgers’ equation with convolution type weak nonlinearity for the Sturm–Liouville operator. We prove that this problem has an explicit solution in the form of series. To achieve our goals, we use methods that correspond to various fields of mathematics, such as the theory of partial differential equations, mathematical physics, and functional analysis. In particular, we use the Fourier analysis method to establish the existence of solutions to this problem on the Sobolev space. As far as we know, it is the first result obtained for the convolution type Burgers’ equation. Since, we use the Fourier analysis method we gave the properties of Fourier transform when acting on convolution, and also gave a property of fractional order of the Sturm–Liouville operator. The generalized solutions of the convolution type weak nonlinear Burgers’ equation with the initial Cauchy condition are studied.