A Study on Solutions of Some Convection Diffusion Equations

Abstract

The Burgers’ equation ut + uux = νuxx is a nonlinear partial differential equa- tion, named after the great physicist Johannes Martinus Burgers (1895-1981). Our study mainly focuses on (global) weak solutions of Cauchy problem for non- homogeneous Burgers’ equation with a time dependent reaction term involving Dirac delta measure and their large time asymptotic analysis. Using Cole-Hopf transformation, we consider the associated two initial-boundary value problems by assuming a common boundary along positive t-axis. The existence and unique- ness of the boundary function along that boundary are established with the help of Abel’s integral equation of first kind. Explicit representation of the boundary function is derived. The solutions of associated initial boundary value problems converge uniformly to a nonzero constant on compact sets as t approaches ∞. Also, using this results, the asymptotic behavior of Burgers’ equation is discussed. Secondly, In chapter 3, we consider a Riemann problem for a de-coupled system with locally integrable general source term and obtain explicit solutions. We find the weak solutions by the method of characteristics. Then we find the shock waves involving delta measures. Also, rarefaction wave solution is derived. In chapter 4, the heat equation with Heaviside function in the source term equipped with Neumann boundary conditions and cosine function as the initial data is considered. In the first part of the article, we are focused to study corre- sponding two initial-boundary value problems and existence of the derivative of boundary function introduced along positive t-axis due to unit step function. The existence and uniqueness of the same is shown with the help of Volterra’s integral equation of first kind. Also, we are concerned with large time behavior of the solutions to associated initial-boundary value problems.