Is the Solution to Parabolic PDEs Always of the Form u = xf_1(φ) + f_2(φ)?

[jbusc]

Hi,

when solving PDE's of the form $$au_{xx} + 2bu_{xy} + cu_{yy} = 0$$ where $$ac - b^2 = 0$$ (i.e., parabolic)

is the solution always of the form:

$$u = xf_1 (\phi) + f_2(\phi)$$

where

$$\phi$$ is the solution to the characteristic equation $$a(y')^2 -2by' + c = 0$$

If not, is there a general form in this sense? (Related to the heat equation in the same way that d'Alembert's form relates to the wave equation)

Thanks, any help at all please is welcome.

 

[gtoguy97]

Yes, the general solution to a parabolic partial differential equation of the form au_{xx} + 2bu_{xy} + cu_{yy} = 0 is always of the form u = xf_1 (\phi) + f_2(\phi), where \phi is the solution to the characteristic equation a(y')^2 -2by' + c = 0. This form of the solution is often referred to as the d'Alembert's form, and it is related to the heat equation in the same way that d'Alembert's form relates to the wave equation.

 

[tacman]

The method of characteristics is a powerful technique used to solve partial differential equations (PDEs) of the form mentioned in your question. It is commonly used for parabolic PDEs, which involve second-order derivatives in one independent variable and first-order derivatives in the other independent variables.

To answer your question, yes, the solution to a parabolic PDE of the form au_{xx} + 2bu_{xy} + cu_{yy} = 0 is always of the form u = xf_1 (\phi) + f_2(\phi), where \phi is the solution to the characteristic equation a(y')^2 -2by' + c = 0. This is known as the characteristic form of the solution.

In general, the method of characteristics works by transforming the original PDE into a system of ordinary differential equations (ODEs) along the characteristic curves. These characteristic curves are defined by the characteristic equation, and the solution to the PDE is then given by the solution to the ODE system.

In the case of parabolic PDEs, the characteristic equation has a unique solution \phi, and the characteristic curves are given by the equation x = \phi(y). This means that the solution to the PDE can be written in terms of \phi, which is why it takes the form u = xf_1 (\phi) + f_2(\phi).

In terms of the heat equation, the method of characteristics provides a general solution in the form of a superposition of two functions, similar to how d'Alembert's form provides a general solution for the wave equation. However, the specific form of the solution will depend on the initial and boundary conditions of the problem.

I hope this helps clarify the use of the method of characteristics for parabolic PDEs. It is a powerful tool for solving these types of equations, and I encourage you to explore it further.