Solving PDE: How to go about it?

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In summary, the conversation discusses how to solve a PDE with the equation 2\frac{\partial^2X}{\partial a \partial b} + \frac{\partial X}{\partial a}(x^4-1) = 0 and how to approach it with X being a function of three variables, x, a, and b. It is suggested to let U=\partial X/\partial b and the problem simplifies to 2\frac{\partial U}{\partial a}+ (x^4- 1)U= 0. This can be solved as a separable equation and then integrated again with the "constant of integration" being an arbitrary differentiable function of a and x. Later
  • #1
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Homework Statement



[tex] 2\frac{\partial^2X}{\partial a \partial b} + \frac{\partial X}{\partial a}(x^4-1) = 0 [/tex]

Homework Equations



How do I go about solving this PDE ??

The Attempt at a Solution



Please help !
 
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  • #2


what is X a function of X = X(a,b,x)?
 
  • #3


If you have written that correctly and X really is a function of the three variables x, a, and b, then let [itex]U= \partial X/\partial b[/itex].

The problem becomes
[tex]2\frac{\partial U}{\partial a}+ (x^4- 1)U= 0[/tex]

which, since we have differentiation with respect to a only, is the same as
[tex]2\frac{dU}{da}= (1- x^4)U[/tex]
where we are treating x and b as constants. This is a separable equation:
[tex]2\frac{dU}{U}= (1- x^4)da[/tex]

[tex]2ln(U)= (1- x^4)a+f(b, x)[/tex]
[tex]\frac{dX}{db}= U= F(b, x)e^{((1-x^4)a)/2}[/tex]
(F(b,x) is [itex]e^{f(b,x)}[/itex] and is simply an arbitrary differentiable function of b and x.


Now, just integrate again. Your "constant of integration" will be an arbitrary differentiable function of a and x.
 
  • #4


There was a mistake in what I originally posted. The PDE to be solved is simpler:

[tex] \frac{\partial^2X}{\partial a^2} + (X^4-1)\frac{\partial X}{\partial a} = 0 [/tex]Would really appreciate your input. Thanks
 
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  • #5


so now X = X(a) and it becomes a nonlinear ordinary differential equation, not a partial?
 

Related to Solving PDE: How to go about it?

1. What is a PDE and why is it important to solve?

A PDE (Partial Differential Equation) is a mathematical equation that involves multiple variables and their partial derivatives. It is important to solve because it is used to model real-world phenomena in fields such as physics, engineering, and economics.

2. What are the steps involved in solving a PDE?

The steps involved in solving a PDE vary depending on the type of PDE and the desired solution. Generally, the steps include identifying the type of PDE, applying boundary and initial conditions, and using analytical or numerical methods to solve the equation.

3. What are the different types of PDEs?

There are three main types of PDEs: elliptic, parabolic, and hyperbolic. Elliptic PDEs involve steady-state problems, parabolic PDEs involve time-dependent problems, and hyperbolic PDEs involve wave-like problems.

4. What are the analytical methods used to solve PDEs?

Some common analytical methods used to solve PDEs include separation of variables, Fourier series, and Laplace transform. These methods involve breaking down the PDE into simpler equations and solving them individually before combining the solutions.

5. What are the numerical methods used to solve PDEs?

Numerical methods involve using algorithms and computer programs to approximate the solution to a PDE. Some common numerical methods include finite difference methods, finite element methods, and spectral methods.

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