Help: analytical of 2nd order PDE

In summary, the conversation discusses the method of solving a 2nd order PDE with given initial and boundary conditions. The question is whether self-similar is the only method or if there are other methods available. The speaker also asks how to set the similarity variable if self-similar method is used. The expert summarizer explains the use of separation variable and obtaining two 2nd order ODEs. The speaker then asks how to solve the second ODE.
  • #1
pangyatou
4
0
What method can I use to analytically solve the following 2nd order PDE?
u=u(x,t)
∂u/∂t - a*x*∂u/∂x-D*∂^{2}u/∂t^{2} = 0
I.C.: u(x,t=0)=u_i
B.C.: u(x=+∞)=0
u(x=-∞)=1

Is self-similar the only way to solve it, or is there any other method can be used to solve it?
How to set the similarity variable if I use self-similar method?

Thanks
 
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  • #2
I use separation variable:
u(x,t)=F(η)T(t): η=sqrt(a/D) x
Then obtain the following 2 ODEs:
(1)T'+a λ T=0
(2)F"+η F'+λ F=0

How to solve the second ODE?
 

Related to Help: analytical of 2nd order PDE

What is the analytical solution for a second order PDE?

The analytical solution for a second order PDE is a closed-form expression that satisfies the PDE and its boundary conditions. This solution involves using mathematical techniques, such as separation of variables or the method of characteristics, to obtain an exact solution.

What are the main differences between first and second order PDEs?

First order PDEs involve only first derivatives of the dependent variable, while second order PDEs involve second derivatives. This means that second order PDEs are more complex and may require additional techniques, such as Fourier series or Laplace transforms, to solve.

How do you determine the type of a second order PDE?

The type of a second order PDE can be determined by looking at the highest order derivative present in the equation. If the highest order derivative is second order, it is classified as a second order PDE. Additionally, the coefficients of the highest order derivatives can also provide information about the type of the PDE, such as whether it is elliptic, parabolic, or hyperbolic.

What are the boundary conditions for a second order PDE?

The boundary conditions for a second order PDE specify the behavior of the solution at the boundaries of the domain. These conditions can be either Dirichlet boundary conditions, which specify the value of the solution at the boundary, or Neumann boundary conditions, which specify the derivative of the solution at the boundary.

How do you solve a second order PDE numerically?

A second order PDE can be solved numerically using techniques such as finite difference, finite element, or spectral methods. These methods involve discretizing the PDE and solving the resulting system of equations using numerical algorithms. The accuracy of the solution depends on the size of the discretization and the chosen numerical method.

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