How do I solve a PDE of this form

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In summary, the conversation discusses a PDE with coefficients that are functions of t and r, and the desire to solve for two unknown functions, alpha(t,r) and beta(t,r). The addition of a constraint is also mentioned, and the use of Laplace transforms is suggested as a potential solution method.
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a14smith
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Hi:

I have come across a PDE of the following form in my research:

C_1 \alpha(t,r) + C_2 \partial_t \alpha(t,r) + C_3 \partial_r \alpha(t,r) + C_4 \beta(t,r) + C_2 \partial_t \beta(t,r) + C_3 \partial_r \beta(t,r) = 0

where the coefficients C_i are all functions of t and r: C_i = C_i(t,r). I want to solve for the functions \alpha(t,r) \beta(t,r). I understand that this is one PDE for two unknown functions. I guess what I would like to do is solve for \alpha(t,r) in terms of \beta(t,r).

In addition, I would also like to solve it subject to the constraint

A_1 \alpha(t,r) + A_2 \partial_t \alpha(t,r) + A_3 \beta(t,r) + A_4 \partial_r \beta(t,r) = 0

where once again the coefficients A_i are all functions of t and r: A_i = A_i(t,r).

Any suggestions or names of methods used to solve equations or I guess really a system of equations when the constraint is considered would be helpful. Thanks in advanced!
 
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Have you considered using Laplace transforms?
 

FAQ: How do I solve a PDE of this form

How do I solve a PDE of the form ax2 + by2 = c?

The general approach to solving a PDE of this form is to use the method of characteristics. This involves finding a set of curves, known as characteristic curves, that are solutions to the PDE. Then, by finding a function that satisfies the characteristic equations, the general solution to the PDE can be obtained.

What is the difference between a PDE and an ODE?

A PDE (partial differential equation) involves multiple independent variables, while an ODE (ordinary differential equation) only involves one independent variable. This means that the solution to a PDE is a function of multiple variables, while the solution to an ODE is a function of just one variable. Additionally, the derivatives in a PDE are partial derivatives, while the derivatives in an ODE are ordinary derivatives.

Can I use separation of variables to solve any PDE?

No, separation of variables can only be used to solve certain types of PDEs, namely linear, homogeneous PDEs with constant coefficients. It is not a universal method for solving PDEs and other techniques, such as the method of characteristics or numerical methods, may be needed for more complex PDEs.

What boundary conditions do I need to solve a PDE?

Boundary conditions are necessary to solve a PDE, as they provide information about the behavior of the solution at the boundaries of the domain. The specific boundary conditions needed depend on the type of PDE and the problem being solved. Common boundary conditions include specifying the values of the solution at certain points on the boundary or specifying the values of the solution's derivatives at the boundary.

Do I need to know the exact solution to a PDE in order to solve it?

No, it is not necessary to know the exact solution to a PDE in order to solve it. In many cases, an exact solution may not even be possible to find. Instead, approximate solutions can be obtained using numerical methods such as finite difference, finite element, or spectral methods. These methods involve discretizing the PDE and solving the resulting system of equations on a computer.

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