Help: analytical of 2nd order PDE

Click For Summary
SUMMARY

The discussion focuses on analytically solving a second-order partial differential equation (PDE) given by ∂u/∂t - a*x*∂u/∂x - D*∂²u/∂t² = 0, with initial and boundary conditions specified. The user initially employs the method of separation of variables, leading to two ordinary differential equations (ODEs): T' + a λ T = 0 and F'' + η F' + λ F = 0. The main inquiry is whether the self-similar method is the only viable approach for solving this PDE and how to appropriately define the similarity variable.

PREREQUISITES
  • Understanding of second-order partial differential equations (PDEs)
  • Familiarity with separation of variables technique
  • Knowledge of ordinary differential equations (ODEs)
  • Concept of self-similarity in mathematical analysis
NEXT STEPS
  • Research the self-similar method for solving PDEs
  • Study techniques for solving ordinary differential equations, particularly second-order ODEs
  • Explore the method of characteristics for PDEs
  • Learn about similarity variables and their applications in PDE analysis
USEFUL FOR

Mathematicians, physicists, and engineers interested in solving complex second-order partial differential equations and those seeking to deepen their understanding of analytical methods in PDEs.

pangyatou
Messages
4
Reaction score
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
 
Physics news on Phys.org
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?
 

Similar threads

Undergrad 1D time dependent PDE - using orthogonal collocation
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad How to Solve a Nonlinear PDE with Sinh Function?
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Residual of PDEs as convergence criteria of numerical solution
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Generic Solution of a Coupled System of 2nd Order PDEs
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Stuck on solving a non homogenous (linear) PDE
  • · Replies 36 ·
2
Replies
36
Views
5K
Graduate How to separate variables in this PDE?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Determining whether the Dirichlet problem is nonhomogenous
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad How to do Magnus expansion for time-dependent companion matrix?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Finding the time for the first shock for a quasilinear first order PDE
  • · Replies 4 ·
Replies
4
Views
4K
Graduate 2nd Order PDE Using Similarity Method
  • · Replies 5 ·
Replies
5
Views
2K