The ability of a recent formulation of the Tau method of Ortiz and Samara to give approximate solutions of a high accuracy of linear partial differential equations with variable coefficients is used to produce numerical solutions of nonlinear partial differential equations. Examples given in this paper show that even for relatively low degrees, Tau approximations give a high degree of accuracy. Besides, the approximate solution and all its derivatives are continuous in the domain.

Dear Ali,

    Thanks for contribution, but where is the paper you talk about?

Regards, Khaled

You must specify the initial and boundary conditions.

Yes, Khaled, there are perturbation methods. But the form of seeking solution can be very specific. You have to try different. Like series over powers of small parameter epsilon, then collecting terms of the higher order, then order epsilon, etc.

I did such derivations for more complex systems in the past; see expansion (6) in https://www.researchgate.net/publication/275581998_Evolution_of_long_nonlinear_waves_on_shelves

About the core equation. The terms P_rr +(2/r)P_r - \lambda P reminds me Laplace operator in polar coordinates if we have angular symmetry; see https://www.math.ucdavis.edu/~saito/courses/21C.w11/polar-lap.pdf .

Here your equation is ordinary. Where is the second variable to make it PDE? The temporal variable is only in given functions, but not in the derivative. Can you present original physical model that gave you this?

Here http://www2.ph.ed.ac.uk/~paboyle/Teaching/PhysicalMaths/Notes_2010/notes_2010_part3.pdf you can see equation (6.4) that reminds even more your type. I saw that your project is about bubbles in blood, but what is the original physical system there to model their evolution?

P.S. Correctness of formulation to give unique solution may be not a key point here. Asymptotic behavior at infinity (limit) is one of boundary conditions. Don't we need the second, like P(r) -> 0 at infinity too?

Conference Paper Evolution of long nonlinear waves on shelves

This is a strange equation! I'll make a few comments first.

1.  There is no time-derivative and therefore the coefficients which are functions of time merely introduce a passive time-dependency to the solutions. Time is merely a parameter here.

2. The large-r boundary condition, P'->0 (where primes denote derivatives with respect to r), means that P->constant. Unless there is something strange about the approach to this constant, then the constant is f_2(t).  This means that the substitution, P=Q+f_2(t) reduces the original equation to a homogeneous one for Q.

3. Most seriously, when epsilon is nonzero, then the equation has a singular irregular point at r=0.  A general technique is to use Q=e^{S(r)} and to find an ode to solve for S(r). This may be impossible to do analytically, although a large-r analysis is more likely to be possible but intricate.

4. If this equation arises from some physical system, then it is likely that the missing BC is that P (or Q) is finite at r=0.

What follows next will be determined by the complexity (and non-impossibility) of solving for S(r). This detail varies from cases to case.

However, if lambda>0, then I suspect that the two Complementary functions for P will have one growing exponentially (or superexponentially) and the other decaying. If the latter is also singular at the origin, then Q=0 or, equivalently, P=f_2(t), is the final solution for all values of epsilon.

Some further thoughts....

Consider the solution when r0. The other complementary function is a constant. On the other hand, if a0. This means that it cannot be part of our solution. So one can say that, if this ODE is being solving in 0

The core equation (i.e. with epsilon=0), P''+(2/r)P'-lambda*P is the modified spherical Bessel equation of order zero. Sounds frightening, possibly because it is!

Andrew, why are you frightened by Bessel equation? Engineers and physicists use it a lot. Bessel functions are tabulated and I saw their graphs 40 years ago. Some are singular in zero, but what is the problem if we know asymptotics? Yes, not good for numerical analysis (machine can break at singularity), but OK for mathematicians and theoretical physicists.

Klaled, I saw your presentation https://www.researchgate.net/publication/316122845_Evolution_of_Gas_Bubble_Growth_in_vivo_at_Constant_Ambient_Pressure

There you have a system with analytical solution. Radius of bubble in blood grows with some time law. But perhaps something happens well before infinity, when its size becomes equal to width of blood vessel. If you would explain its meaning to us and modification to get your new equation, it can be useful for discussion.

Poster Evolution of Gas Bubble Growth in vivo at Constant Ambient Pressure

Hi Yuri, I was only joking about being scared of Bessel functions!  My humour.....and occasionally that doesn't work That said, I have seen people's faces turn white when I have given them the sad news that further progress in their work can only be made by using them! I have taught Bessel functions to undergraduates in the past and occasionally they turn up in my research. My view of them is just like having to repair a bicycle tyre miles when from home - I need to do it, I know how to do it, it takes up too much time, and I would rather not have to do it! However, success brings its reward.

Oh yes, Andrew, now I see that it was humour. But since many faces turn pale - not humour for all! Well, there are many special functions, but at least with these Bessels you can have orthogonal set and can formally find solutions for Laplace and wave equation in spherical coordinates. In the past a lot of applied research was on equations of mathematical physics on some nice surfaces, like sphere and cylinder. I even remember that there are about 10 systems of coordinates where variables can be locally orthogonal and thus separated. More interestingly (perhaps for you) is that I used one of such exotic systems (that even not all physicists use) with economic application - elliptic-hyperbolic coordinates - to calculate the demand change in two-dimensional Hotelling model: https://www.researchgate.net/publication/228250923_Hotelling%27s_Revival_Demand_Continuity_in_R2_for_a_Large_Class_of_Functions

The orthogonal set was between ellipses and hyperbolas! The set of indifferent consumers to buy from two firms (when transport costs are accounted in 2-dimensional space) lies on hyperbola. Thus, the change of demand from one firm if it changes price is the integral between two hyperbolas, that can be calculated as 2-dimensional integral even with non-uniform density.

Article Hotelling's Revival: Demand Continuity in R2 for a Large Cla...

Ah, but one can obtain complete sets of orthogonal functions for certain second order equations - Sturm-Liouville equations. I am pretty sure that that is the case for standard Bessel Functions, J_n, and for their spherical counterparts, but the modified ones tend to produce curves that either grow or decay monotonically, either exponentially or super-exponentially.

Your application sounds very interesting. I have also used Bessel functions in various convective instability problems, and in a 2D filter for analyzing the characteristics of atmospheric waves.

At one point I played around with a Fourier Series composed of Airy functions. Nothing has been published from that, unfortunately, although it too will have a complete orthonormal series associated with them. I am guessing that a Fourier-Airy transform might also exist as an analogue to the Mellin transform.

I agree with the previous comment - try numerical methods. An analytical solution may not be possible.

The initial and boundary conditions need to be specified - are they known?

Look for the solution as an infinite power series. You have to add initial and boundary conditions.

For epsilon taking nonzero values, there won't be a power series solution if the inner boundary is at r=0; the problem has an irregular singular point there.

Standard numerical methods will have problems close to r=0 because of the 1/r^2 term. That's not to suggest that a numerical solution is impossible.

However, it remains for us to know whether the inner boundary condition is at r=0 or not. If not, then a numerical solution may well be straightforward.

At this stage, P=constant does satisfy the equation!

If I understand your question correctly, you do not have a PDE -- you have a quasi-steady state problem.  This is a family of radial ODEs (independent of any angular component, but parametrized by t) in polar coordinates for the exterior of a disk of radius R.  Here, R is the "inner boundary"  and I would assume R>0.  

   Without having worked it out, I think the solution P(r,t)=f_3(t) is the unique solution (although this uniqueness may depend on \lambda). 

   I thought I had answered this and came back to correct an error I noticed in what I had written earlier -- which apparently never was distributed.

   The point is that this is not a PDE at all, but a family of ODEs (ignoring any angular dependence but taking this as parametrized by t) for a quasi-steady state problem in polar coordinates for the exterior of a disk of radius R.  Solving this for each t by ODE methods should be fairly straightforward.  [I do note that P=constant is not a solution unless \lambda=0 or f_3=f_2.]

Thomas, yes you are right with you parenthetical remark. Oops! Many thanks.

It's also a linear ODE. I still stand by my comment that the irregular singular point at r=0 ma cause numerical problems.

Great comment , prof Thomas.

Best regards.

Cher Khaled

J'ai consulté les commentaires des collègues et je suis d'accord avec celui du Prof. Thomas. Il n'y a pas de dérivée en t, il s'agit donc bien d'une famille paramétrisée d'EDO linéaires, du second ordre, pour lesquelles je ne connais pas de solution analytique, sauf cas particulier, Je suggère donc aussi une approche numérique.

Meilleurs sentiments

J.-A. Marti

Cher Khaled

A la réflexion je fais l'hypothèse que tu souhaites avoir une famille paramétrée par epsilonn de solutions en (t,x) variables. Et la prendre comme représentant d'une solution généralisée dans une algèbre convenable de type Gevrey ou autre. Est-ce bien cela?

Dans ce cas il te faudrait  une solution explicite et non numérique. Peux-tu alors te référer aux suggestions de Youri Egorov sur ce sujet qu'il connait bien?

Amitiés

Jean-André Marti

Dear colleagues

Thank you very much for your valuable contributions.

My best regards

Khaled

Chers collègues

Merci beaucoup pour vos précieuses contributions.

Mes meilleures salutations

Khaled

Dear Abderrahim,

 To solve non linear PDE you have the choice between two efficient methods :

The Newton method and the fixed point method. You can have a look on my last publication for solving 3D température equation with variable conductivity.

Best regards,

I think this problem can be solved using 'Galerkin method + Perturbation methods' , converting PDE into ODE and then apply Perturbation methods. I think, I have come across problems in acoustics solved using 'Infinite Elements Methods', you can take a look at it.  Further, you can always go for asymptotic methods in the limit of large or small epsilon to see what happens to your 'P(r,t)'.  I hope this may give some idea.

Regards

Balasundaram M

Dear Khaled,

I have had an attempt at solving your problem, although I am a little rusty on this, and I obtained an interesting solution!

This looked a little bit like the sinc (cardinal sine) function shifted upwards along the 'y-axis', dependent upon certain parameter values.  Its pressure gradually reached a steady state value, P > 0 as t approaches infinity (as far as I can remember), but this was also dependent upon the initial function of time, G(t) (or F(t)), which may well be different from the one that you have (or would choose to substitute).

I chose the decreasing exponential function, G(t) = Exp(-ct), to test out this solution.  I treated the problem as a 'singular point', in which the function G(t) (or the related  F(t)), was merely a parameter, but the solution that I eventually obtained involved a final summation term that resulted in the hyperbolic sine function, or sinh.  You may obtain something entirely different if you chose a different function for G(t) (or F(t), as the case may be), so it all depends upon your model and what your function is.

My question is: would you be interested in seeing my worked, hand-written, answer of around 7 pages, if you think that it will be of any use?  I don't know if it will help you but, so far, it is currently of no use to me, so do let me know?  I am afraid that I just like to 'mathematically doodle', and work out complex problems for the fun of it!

I look forward to your response.

Kind regards,

Stephen Mason

Thank you my brother prof @slim kaddeche

Dear Professor Wakif (Message also intended for Khaled G. Mohamed),

I don't know if your last comment was meant for me, but I attach my worked solution(s) below, which I hope is/are helpful.

Please do let me know if they are of use to you.

Kind regards,

Stephen Mason 

Sorry, I made a mistake in the graphical plots in the last attachment by failing to correct a minus sign (which should have been a '+') in the resulting solution, for the purposes of my own particular example.  In this case, I feel that my corrections are more fitting in this respect, given the functions and parameters that I have adopted.  Also, I wanted to mention that I guessed/assumed the nature of the initial conditions in order to arrive at the answers as shown.  This solution will obviously be quite different if other functions, parameters and initial conditions, as befitting the actual physical situation, are used.  This is assuming that my original method of solution of the initial PDE is correct in the first place.

Stephen

Thank you dear prof.Stephen Mason

No problem, Professor Wakif.  I was happy to help out, and hope that my solution was useful.  Cheers for now, Stephen

Dear Professors Mohamed and Wakif,

I wanted to try to satisfy the challenge, at least for myself, of finding a general solution to your PDE problem for which epsilon is non-zero.  Having been quite rusty on this theory, I had to do quite a lot of reading around, but finally achieved the objective, I hope. Please find attached my hand-written notes for the general solution to your original PDE, now including the epsilon parameter.  I should make you aware that Mathematica also returns the same answer that I obtained independently, so I feel confident that I am at least on the right lines of enquiry.  If I can be of any further assistance, then please do let me know.

Kind regards,

Stephen

This is a linear partial differential equation. There is no nonlinear terms. Il you send me the expressions of the constants lambda and epsilon, as well as the expressions of the functions f1, f2 and f3 I will send you a code based on spectral collocation method to solve this equation.

Best regards. 

Dear all colleagues,

Thank you very much for your interest, I will put some details for this problem, which I did not prefer to mention it first, because it may disturb the non-specialists in my field of study since this is problem has applications in diving science and some terms may not be understood easily by non-specialists in bubble dynamics and physiology sciences. But I can say in brief, this system represents a boundary value problem with moving boundary. I will give some details as soon as possible whenever I have enough time to do so. I am sorry I am so busy these days.

My best regards,

Yours Khaled.

Take your time Pr. Mohamed.

I'm waiting to hear from you soon.

Best regards.

Dear Professor Mohamed,

Thank you for your response, and I too am very much looking forward to viewing the further information that you have regarding this problem.

Kind regards,

Stephen

You should try COMSOL Multiphysics wich allows to solve user-edited equations. Bets regards, Floricica

It is indeed a linear equation, I recommend singular perturbation expansion, however r=0 at the O(eps) solution must be analyzed carefully.

try it on adomian decomposition method

try homopy analysis method

I think you may use Comsol Multiphysics for solving this problem which is based on Galerkin Weighted Residual Method. After converting PDE into ODE, you can apply Perturbation methods.

Regards

Md. Mojammel Haque

This PDE could potentially be studied through Lie symmetry analysis.