A Homotopy Method For Solving One-dimensional Convection Diffusion Equations

In this paper, we mainly study the homotopy method to research the dynamic behavior of the one-dimensional convection diffusion equation. The convection diffusion equation has been widely used. From the spread of various pollutants in the atmosphere and the water emissioned by human to the absorption of drugs, from the seepage tracking in the porous medium to the spread of solubles matters in estuaries and coastal waters, from the temperature of the atmosphere to the heat-transfer in closed containers,these are closely related to the convection and diffusion. At the same time, the homotopy analysis method is one of the most effective methods in solving nonlinear problems. It is connecting a nontrivial problem (easy to get solution) to the original question (nonlinear problem) into a series of linear problems to solve by constructing homotopy equation. This method is applied both weak and strong nonlinear problems. Therefore, it has important theoretical significance and practical value to study the homotopy analysis method for solving convection diffusion equation.For the one-dimensional Burgers’equation which with initial-boundary conditions, we take the linear part of the Burgers’equation as the auxiliary linear operator and construct a new homotopy equation. The homotopy approximation solution which we get can accurately capture shock wave and reflect the true physical phenomenon. It is better in numerical approximation accuracy compared with other results.For the one-dimensional Burgers-Huxley equation which with initial-boundary conditions, we select the appropriate operator and construct a homotopy equation. The homotopy approximation solution is not only having widely effective area, but it can accurately capture shock wave and handle very sharp boundary layers.