A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

Abstract

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

Effects of Nitrogen Fertilizer Ratio on Microbial Diversity of Rice Rhizosphere Soil in Cold Region

“…In order to meet the needs of social and economic development and the growing population for food, increasing the application of nitrogen fertilizer is one of the indispensable means (11). However, excessive input of nitrogen fertilizer will cause a large amount of waste of fertilizer nitrogen and economic losses but also lead to environmental quality deterioration and decline in the quality of agricultural products (12).…”