1. Could you perhaps educate us on what the SLM method is?

2. This looks very much like a course exercise problem, which you are supposed to do on your own. Can you convince us that this is not the case? 

this is not problem of exercise but this new developing method author is P.Sibanda and you may ref.    Thsi is  Mathematical model in fluid flow problem.This is for kind information to Kare Olaussen

Your differential equation is implicit and thus subject to singularities. In fact, your left boundary condition is at a singular point. This makes the application of any solution method - no matter whether analytic or numerical - non-trivial, as already the question of existence is highly non-trivial. I am not aware of existence results for boundary value problems for implicit equations. But in the case of initial value problems anything may happen at singular points from no solution over finitely many to infinitely many ones. Therefore I would suggest to start with a theoretical analysis of the nature of the singularity at the left boundary and only afterwards consider any approximation scheme. In particular, from the little bit I have seen on SLM I am not sure whether it is a good method to approach your problem. The linearisation will remove the singularity and thus drastically change the character of the problem. Therefore it appears highly unlikely that you will observe a convergence to a solution.

I am very interested in such implicit differential equations. Can you give me some more details where your equation come from?  

Thank you Kare Claussen and Werner M. Seiler sir for your interest in my problem. Werner sir, I am attaching some more details with this for your reference and hoping that you can understand and help me to resolve it.

Bhumika.

I didn't look at your problem again before now. I couldn't make much progress. I note that there are two exact solutions, both fulfilling one of the boundary conditions,

y = 1 and y = -2x^2/3

I looked at ways to generalize/perturb the last solution, but found none! Do the physics of your problem require y to be positive?

Sorry, my above solution was not to the problem you attached to your question (but to the similar problem in the paper you linked to).

For the original problem there is still a special exact solution, y = K x^2, with K=(A-B)/6.

More physical insight can be gained by introducing Y = y^2/2, and rewriting the equation as

Y'' + L sqrt(Y) = 0, where  L = \sqrt(2)( B/2 x d/dx - A).

This is Newton's equation for a particle in a potential V ~ -Y^(3/2), with some time dependent friction, starting at Y=0 at time x=0. You just have to give it sufficient initial velocity Y' for it to reach Y=1/2 at time x=1. There should be no difficulty solving this problem numerically by the shooting method, using any standard ODE solver. It seems that there may be two solutions to your problem when A-B = 6.

For B=0 the equation has a conserved energy, and can be integrated exactly (providing  x as a hypergeometric function of Y --- I don't think that relation can be inverted). For a series expansion of the generic solution at low x, it seems useful  to introduce Z = Y/x, and a new "time"variable u = x^(3/2). That should make it possible to express Z as a power series in u (whether that series has a finite radius of convergence is another matter).

Dear Bhumika!

Thanks for the file with details with your model! So you are looking at a similarity reduction of the porous medium equation in the case of slow diffusion - which leads to your singularity. Let me explain what is particular about your equation and why you are probably experiencing problems in your computations.

An explicit second order ODE (like it is treated in most textbooks) has the form y''=f(x,y,y'). In the case of an initial value problem you impose two initial conditions, say y(0)=y0 and y'(0)=y'0 which allow you to determine y''(0)=f(0,y0,y'0). Differentiating repeatedly your equation you can then produce without problems a Taylor approximation of your unique solution (under mild conditions on f). Alternatively, you impose two boundary conditions like y(0)=y0 and y(1)=y1.

Now your equation is not explicitly solved for y'' but of the form yy''=f(x,y,y') and one of your boundary conditions is y(0)=0. If you enter it into your equation, then y'' drops out and you find that you must have y'(0)=0 - even if you wanted to study an initial value problem you could not impose a second initial condition! Hence it is not possible to compute in the standard way a Taylor approximation and it is also not possible to use a shooting method to solve numerically your problem. I also strongly suspect that this fact leads to your problems with your approximation method.

A closer inspection of your left boundary condition shows that you are facing a so-called irregular singularity - which is the harder case. I have not had yet the time to make a more detailed analysis of what is going on there. In principle, there are essentially three possibility (still with only taking this one boundary condition into account!): there exist zero, two or infinitely many solutions. The first case we can exclude, as y(x)=0 is obviously a solution (albeit one which does not satisfy your other boundary condition). In the second case you can only hope that the second solution also satisfy the other boundary condition and I do not see any simple way to check this. In the last case it could happen that one (or several!) of the solutions satisfy the second condition. Under certain circumstances it might here be possible to apply some variation of the usual shooting method in order to construct numerically such a solution. But all this depends on a more detailed knowledge of the type of singularity your model features. If I find some time, I will try to dig a bit deeper.

I am more a pure mathematician and I have not yet seen any theory on boundary value problems with a condition imposed at an irregular singularity. It might be that applied mathematicians have solved some problems of this kind by ad hoc methods, but I am only aware of some articles studying initial value problems. In any case I doubt very much that your approximation method will work in such a situation. Such expansions always implicitly assume the usual existence and uniqueness theorems for ODEs. However, these theorems fail at singularities. The typical behaviour at irregular singularities is that several solutions intersect - something that is not possible for an explicit ODE. Thus I cannot see how such an expansion could produce a solution.

Bhumika,

I took another look at the paper you linked to: The boundary conditions (2) and (5) are not consistent with each other. Your original problem does not admit a similarity solution, due to the boundary conditions.

Moreover, you probably want to provide some initial values, Sw(L,0). Equations (1-2) have a solution Sw(L,T) = sqrt(L). Perhaps any initial condition will approach this solution? 

In Bhumica prob.You may choose y(0)= y0 & dy/dx =0,atx =0 and y(1)=1.

It is not singularities in problem because x is not multiplied to highest derivative term in equation.

Manoj> In Bhumica prob.You may choose y(0)= y0 & dy/dx =0, at x =0 and y(1)=1.

Then you impose too many boundary conditions. For instance, if A=0 the conditions at x=0 imply y(x) = y0. So then you must set y0 = 1 to satisfy y(1) = 1. I.e., you cannot impose the value of dy/dx at x=0, if you fix the values y(0) and y(1).

Now, as I said in an earlier post, properly rewritten this is just a simple mechanical problem of a particle in a potential, with some friction. There are no serious singularities for y(0) >= 0. I have not investigated the case of y(0) < 0, but I think that would be inconsistent with the condition y(1)=1. The problem has an extra "self-accelerating" solution when y(0)=0, because you then start at an unstable equilibrium point. Hence, also that solution is not very exotic from a physical point of view.