There are many methods for such a reduction. Most are described as using some special ansatz like for example travelling wave solutions. However, almost all of these methods can be viewed as a special case of a symmetry reduction (potentially with respect to a non-classical symmetry). For a first introduction to symmetry methods, you may take a look at the books by Olver (Applications of Lie Groups to Differential Equations) or by Bluman and Kumei (Symmetries and Differential Equations). There exist by now many textbooks on this topic for different audiences, but these are two classicas.

You can convert a PDE  into a set of ODEs by spatial discretization using e.g. finite difference methods. This is quite intuitive and good for simulation purposes.

A point of correction: finite differencing yields a set of algebraic equations, not ODEs. One of the most widely-used methods is "Separation of Variables"  (see, for example "Heat Conduction" by MN Ozisik), which you often see applied to PDEs having a Laplacian operator (e.g. for thermal diffusion in 2 or 3 dimensions). As Werner mentioned, symmetry reduction is also a possibility for certain systems governed by a PDE. This often works for the PDEs in thermofluids when there are no obvious length or time scales. Wikipedia has a good intro at https://en.wikipedia.org/wiki/Similarity_solution but the technical literature on this topic is enormous.

Christoph is right, as a semi-discretisation of a PDE indeed yields ODEs (or more generally a DAE). Here one may use all kinds of numerical approaches like finite differences, finite elements or spectral methods but also integral transforms. However, the solutions of the thus obtained ODEs are only approximations of solutions of the original PDE. By contrast, symmetry methods lead to exact solutions. Here the question is what Nikunj precisely means by "conversion". Separation of variables is by the way also only a special case of a (non-classical) symmetry reduction. I have not yet seen any reduction method that could not be interpreted as a symmetry reduction. Of course, it is another question whether one should do this: you can easily do a separation of variables without having ever heard of symmetries.

You can use the Fourier transform (for evolution problem)

You could try the Galerkin method. It is used for discretising continuous problems.

There are several methods depending on the nature of the PDE (linear or nonlinear) and what you want to do with the ODE obtain. As already mention above Galerkin method is good for non-linear PDE in infinite dimensional spaces.you can also use  it in for linear case if you want numerical solutions. Another method is the  Laplace transform if you are in a  finite dimensional space (Fourier transform) .You can also check  #Text book: principles of applied mathematics(Transformation and approximation) by  James P.keener#

For two variable PDE's problems you can use Laplace transform methods. For three variables PDE's you can use Laplace and Hankel transforms methods.

@Werner. I agree that semi-discretization, or what is commonly called the "method of lines" is indeed another possibility. Christoph mentioned finite-differencing, which almost always means recasting the PDE(s) into algebraic equations --- we probably just have semantic differences here. The issue is basically the same with the Galerkin method, mentioned several times above. I would add that if the original poser of the question is only interested in a list of such methods, then what has been posted by the commentators here is a pretty good start. Conversely, he seems to be looking for something to "solve any problem of any section in mechanics", which is a much different proposition. The properties vary among these methods and picking the "best" one for any given problem is not a trivial task, for example if using Laplace Transforms, it is often very difficult to obtain the inversion rule to recover the solution.

Some methods:

1-finite difference method

2-finite element method

3-homotopy perturbation method

4-homotopy analysis method

5-Adomian decomposition method

6-variational iteration method

7-shooting method.

There are many methods.

 Perturbation method, Numerical methods. Even this can be done using the transforms method.

This depends on what you mean by "convert".   There are methods in which ODEs are used in solving PDEs (as separation of variables to get a series) and approximation methods (e.g., method of lines with semi-discretization) used for computation.  As Yousif mentioned, there are homotopy methods, typically assuming one can solve some class of simpler PDEs and use ODEs with respect to a parameter to connect these.  I would add to this, primarily for theoretical purposes, the use of semigroup theory to view a PDE as an ODE in an infinite dimensional state space.

Of course, there are many methods ... By the moment please see https://www.youtube.com/watch?v=JFnlX5Pcjcg

You present one of the partial derivatives on the formula of finite differences. You will have a system of ordinary differential equations. A count of equations in the system will depend of step.

There is also, the Legendre wavelet method to convert a partial differential equations to ordinary ones.

I note that the video suggested by Florian Munteanu describes (a special case of) similarity solutions.  These are related to deep properties of the PDEs (and of the underlying physics) in considering structure preserving transformations: symmetries leaving the PDE invariant).  I note as particularly important the search for traveling wave solutions leading to d'Alembert's solution of the wave equation and, for a nonlinear example, to solitons.

Hello Nikunj...

There are several methods. You can use Transform Functions such as Laplace Transform for 2 variable and Laplace and Henkel Transform for 3 variable non homogeneous PDEs and using inverse Laplace Transform you can get the analytical expressions. As far as variable separation method is concerned one can use only for homogeneous linear PDEs. Again there are other methods such as finite Difference Method but that will yield a set of algebric equations which can be solved easily but good for simulation purpose. These are good when you don't require any analytical expressions of the PDE you have. I think there is another Cauchy method...Not sure. I hope this will help you to some extent.

Werner M. Seiler , Is the transformation used in different methods e.g. first integral method, exponential method, where the PDEs reduces to odes. Is that transformation is non classical symmetry approach ? any reference which explain the history or origin of that particular transformation.

Method of Lines also convert PDE to ODE by truncating.

One can take the help of the following articles and references therein

https://link.springer.com/chapter/10.1007%2F0-387-30260-3_6

http://www.znaturforsch.com/s65a/s65a0896.pdf