Is the similarity transformation the sole way of transforming the boundary layer governing PDEs into ODE ?Or do we have any alternative ways of doing it??
It depends on what you are trying to do. Which PDE are you hoping to solve? The successfulness and efficiency of various methods depends on the PDE and also the application. Please be more specific.
Dudley J Benton I would like to know if Prandtl's boundary layer equations can only be simplified by using similarity transformation or is there any alternative way of simplifying it?
Yes, similarity transformation, Linearization and Lie Algebra are commenly used techniques and for high dimensional integrals, you may use Laplace approximation or special hypergeometric functions.
Muhammad Ali so that means Prandtl's boundary layer equations can be solved by using either linearization or by lie algebra?
Dear Madiha Takreem,
Linearization is also a way of trnasformation from high order to low order, some people use the dimentionality reduction, localization, laplace transform, wavelet transform etc... Here are some related explers related to your query:
Article A Linearized Model for Boundary Layer Equations
https://www.sciencedirect.com/science/article/abs/pii/S073519331100042X
Article On linearization method to MHD boundary layer convective hea...
Transform a PDE into a system of ODE depends on ability to decouple the variables. For example axial symmetry which can be used to transform the equation of the Hydrogen into a the Strum-Louisville in ODE. The ability to do this is equation and boundary or initial value specific.
The following references shows how one can use various transforms to aid in solving specialized PDE's by transforming into problems that can be solved by ODE's. The Fourier transform is one.
https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.11%3A_Transforms_and_Partial_Differential_Equations
Of course separation of variables is a well known technique when the variables can be decoupled. In the analysis of the Hydrogen atom, one can use the fact that the potential is spherically symmetric (only depends on the distance from the proton) and the problem can be transformed into the the analysis of the equivalent Strum-Louisville problem in ODE.
For first order PDE, the method of characteristics can be used to generate an equivalent set of PDE's. For a rigid body in three space, the dynamics is described by ODE instead of PDE since the dynamics is invariant under SO(3), the Lie Group of rotations in R^3.
In general symmetry that results by the invariance of a Lie Group action the PDE will lead to the decoupling and/or reduction of variables. This can often head to solution by ODE. However, it is very equation dependent.
http://uu.diva-portal.org/smash/get/diva2:1471770/FULLTEXT01.pdf
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