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

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In summary, the method of characteristics is a mathematical technique used to solve for solutions in partial differential equations that depend on both space and time variables. It is typically used for first-order equations in fields like fluid mechanics and thermodynamics, as well as in problems involving wave equations and transport phenomena. The steps involved include identifying the equation and boundary conditions, finding characteristic curves, using initial conditions to determine constant values, and constructing the solution. Some advantages of this method are its ability to solve non-linear equations, accurately capture discontinuities, and ease of implementation in numerical methods. However, it also has limitations such as not working for all types of equations, difficulty with complex boundary conditions, and inefficiency for high-dimensional systems.
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jbusc
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Hi,

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

is the solution always of the form:

[tex]u = xf_1 (\phi) + f_2(\phi) [/tex]

where

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

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.
 
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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.
 
  • #3


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.
 

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

What is the Method of Characteristics?

The method of characteristics is a mathematical technique used in partial differential equations to solve for solutions that are dependent on both space and time variables. It involves finding curves, known as characteristics, along which the solution to the equation remains constant.

When is the Method of Characteristics used?

The method of characteristics is typically used when solving for solutions to first-order partial differential equations, particularly in fluid mechanics and thermodynamics. It is also useful for solving for solutions in problems involving wave equations and transport phenomena.

What are the steps involved in the Method of Characteristics?

The steps involved in the method of characteristics are as follows: 1) Identify the partial differential equation and its boundary conditions, 2) Find the characteristic curves by solving the characteristic equations, 3) Use the initial conditions to determine the constant values along the characteristic curves, 4) Construct the solution by combining the characteristic curves and the constant values.

What are the advantages of using the Method of Characteristics?

The method of characteristics has several advantages, including its ability to solve for solutions in non-linear partial differential equations, its accuracy in capturing discontinuities and shocks in solutions, and its ease of implementation in numerical methods. It also allows for the use of initial and boundary conditions, making it a versatile technique for solving various problems.

What are the limitations of the Method of Characteristics?

While the method of characteristics is a powerful technique, it also has some limitations. It may not work for all types of partial differential equations, particularly those with complex boundary conditions. It also requires the determination of characteristic curves, which can be challenging in some cases. Additionally, the method may not be efficient for solving problems with high-dimensional systems.

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