APPROXIMATION OF THE HOPF EQUATION BY LOADED EQUATIONS

Abstract

Purpose. We investigate the methods for reducing first-order partial differential equations with power nonlinearity to loaded equations using the example of the Hopf equation. The solution to the reduced equation is applied to a sequential approximation of the solution to a nonlinear equation by solutions to a linearized equation. Methodology and Approach. Two methods of reduction are proposed. In the first of them, the desired function in the nonlinear term is replaced by its average value for the spatial variable. To solve an auxiliary ordinary differential equation, a second reduction is possible, namely, to an algebraic equation. In the second method, an integral transition is made to the loaded equation. The resulting auxiliary equation is solved using a partial solution to the corresponding differential inequality. Results. The proposed methods of reduction after some additional transformations allow one to obtain initial approximations for starting the iterative process of searching for approximate solutions to a nonlinear problem. The possibility of using partial solutions associated with the differential inequality equation is shown. Theoretical and Practical Implications. We have demonstrated the possibility of applying reduction to loaded equations to find approximate solutions to first-order partial differential equations with power nonlinearity.

Keywords

• nonlinear equation
• reduction to the loaded equation
• Hopf equation
• boundary-value problem
• approximation