Request for  a  hint: What does it mean  exp(a (d/da))? My guess would be  that this stands for the sum of a series built of the powers  (a d/da)^n. If so, then the problem becomes difference-differential equation of infinite order. Is this really what is meant in the question?

Regards, Joachim Domsta

Dear

Most methods for analytically solving PDEs transform them into systems of ODEs

so I have two suggestions for you

First, to have a look at the following book

Partial Differential Equations: Analytical Solution Techniques (Texts in Applied Mathematics, by J. Kevorkian

And then you can do another way in your research like a doing some work by applied numerical methods to find solutions of these equations.

Regards

Taken literally, my interpretation of this equation is that it means

df(u, u*, t)/dt = iK [f(u+1, u*, t) - f(u, u*+1, t)], with K = M(N - O).

where u = ln(x +iy). So it only connects function values defined on a lattice in the (u, u*)-plane. Further progress can be made by making a Fourier transform with respect to (u, u*). This leads to an ordinary first order differential equation in t for each Fourier component.

thanks Dr. Olaussen but what about if the M, N and O are all functions of a a*

i mean the product of a a* is the same procedure you suggested is applied or some thing else please answer me as soon as possible with more details and again i'm very grateful for you thanks for your valuable assistant with my problem.

Ahmed> ... please answer me as soon as possible with more details ...

First, I think the problem is formulated in a very obfuscating notation, which is a serious obstacle for further progress. This should be addressed first:

You say that a = x + iy, so let me start from the (x, y)-coordinates, and introduce (essentially) cylinder coordinates,

x = exp(μ) cos(φ), y = exp(μ) sin(φ).

In these coordinates we find, by a short calculation, that

a* d/da* + a d/da = d/dμ, a* d/da* - a d/da = i d/dφ.

So, the problem you originally formulated becomes

df(μ, φ, t)/dt = iK exp(id/dφ) f(μ, φ, t), with K = M(N - O).

Now aa* = exp(2μ), and μ only enters as a parameter in the above equation. So it does not really matter if M, N, O depends on aa* or not. To proceed, we expand f in a Fourier series,

f(μ, φ, t) = ∑n fn(μ, t) exp(-inφ).

Since exp(-inφ) is an eigenfunction for id/dφ, id/dφ exp(-inφ) = n exp(-inφ), we find

exp(id/dφ) exp(-inφ) = exp(n) exp(-inφ) = exp(-in(φ+i)), such that

exp(id/dφ) f(μ, φ, t) = f(μ, φ+i, t).

This illustrates my previous remark about the relation between the exponential of differential operators and finite translations, although this is not a very useful observation in the present case (unless you are missing the imaginary unit in your equation, which anyway does not look quite right). In this case, it is better to work directly with the Fourier amplitudes, which must satisfy the equation

dfn(μ, t)/dt = iK exp(n) fn(μ, t),

with solution fn(μ, t) = exp(iKt exp(n)) fn(μ, 0).

Now, the way you write your equation only makes sense to me if there is a sign mistake in one of the exponential functions, such that the equation also involves a term exp(d/dμ) fn(μ, t) = fn(μ+1, t). If M, N, O is independent of μ this can also be solved analytically by another Fourier transform with respect to μ. If there is an explicit μ-dependence, you are stuck with a bidiagonal matrix equation which involves all quantities fn(μ0+m, t), for each μ0 in the interval [0, 1), and m=0, ±1, ±2,... . Whether this problem is also analytically solvable depends on explicit details, but in general it is very unlikely.

thanks Prof. Olaussen very much

You are right Prof. the correct formulation for my problem is

d f(a,a*,t) /dt = - i M {N [ exp(c a*(d/da*)) - exp(( c a(d/da)) ] - O [ exp(-c a*(d/da*)) - exp(( -c a(d/da)) ] } f(a,a*,t)

where d denote to partial differentiation, i imaginary =square root of -1 a and a* are two complex parameters where a=x+i y M, N and O are three function of the product a a*, c is a constant f(a,a*,t) function of a, a* and t (time)

i think the methodology is changed or still the same procedure where taking same cylindrical coordinates and applying the chain rule to transform the problem then applied the Fourier transform for both sides of the equation after expand the function f(μ, φ, t) using Fourier series and then lead to ordinary differential equation with t

i'm hope to keep in touch with you.

Thanks very much Prof. Olaussen and i have a number of questions that i'm tried to answer some of them below, please correct my answer, if i'm wrong.

These questions are arise after working on your procedure to solve my problem:

1) why you introduce this transformation:

x = exp(μ) cos(φ), y = exp(μ) sin(φ).

and how you conclude this transform?

is the cylinder coordinates that you assumed used to overcome the problem of variable coefficients in pde? and why assumed μ as a parameter?.

2) i think a* d/da* - a d/da = - i d/dφ with (-) ve sign where i follow your manner.

3) why you set a* d/da* - a d/da ) f(a,a*,t)= exp(a* d/da*) - exp(a d/da) . f(a,a*,t) in your answer?

i think you are trying to take the expansion for the exponential differential operators for the first order and compute the exponential differential operators tow terms between the parenthesis in order to find the operator form that equivalent to these terms in (μ, φ) notation then add all terms again together to produce the exponential deferential operator in the new coordinates (μ, φ) .

4) the solution of the pde df(μ, φ, t)/dt = iK exp(id/dφ) f(μ, φ, t), with K = M(N - O). is by taking the Fourier transformation for both sides of the pde equation.

5) in my case in the final correction of the pde equation that is send to you professor

d f(a,a*,t) /dt = - i M {N [ exp(c a*(d/da*)) - exp(( c a(d/da)) ] - O [ exp(-c a*(d/da*)) - exp(( -c a(d/da)) ] } f(a,a*,t)

M,N,O are explicitly depends on (μ) and its obviously from the above pde equation that we cannot introduced K = M(N - O) and you told me Little about the problem of bi-diagonal matrix equation if M,N and O are explicitly depends on the (μ) such as in my case , So how i solve this problem and how this change my final analytical solution where i have an initial condition for this problem.

i' m very appreciate if you are give me your answer with details.

6) what is the importance of the exponential differential operator can you give me some of identities or it's rules.

with my best regards and wishes, happy new year for you and for your family.

Ahmed> why you introduce this transformation:

x = exp(μ) cos(φ), y = exp(μ) sin(φ).

They corresponds to a = x + iy = exp(μ+iφ) so μ+iφ = ln(a), which was the transformation I mentioned in my first post. The motivation behind this transformation is to arrive at more familiar derivative expressions, like transforming xd/dx to d/ds, when s = ln(x).

Ahmed> i think a* d/da* - a d/da = - i d/dφ

I get the opposite sign; I will upload a some handwritten calculations with more details.

Ahmed> why you set exp(a* d/da* - a d/da ) f(a,a*,t)= exp(a* d/da*) - exp(a d/da) . f(a,a*,t)

I formulated the problem from (wrong) memory, so I ended up solving a different problem. Sorry for that.

Ahmed> the Fourier transformation for both sides of the pde equation

Yes, except that I would not call it a pde. Because it is a difference equation in the (μ, φ)-variables. Fourier transforms diagonalises the d/dμ and d/dφ operators, so functions of these operators can be treated as regular functions. The most important property here is that the exponential of these operators, when acting on functions, generate translations. I illustrated the essential steps in my previous post. You can note consistency with the Taylor expansion formula, but this property has more general validity.

I will upload some handwritten notes later.

Here are the notes

Thanks very much Prof. Olaussen for your interest and I'm study your notes and used it in my problem and still have some questions and ambiguity which try to put it in the following pdf- document.

please answer me as soon as possible with details.

Thanks very much Prof. Olaussen for your interest in the last discussion you relate the further progress to solve my pde with getting more information about the problem and i put all the information you need in the attachment pdf. document that send to your e-mail.

please, answer me as soon as possible.

Dear Prof. Olaussen i send the papers you need to your E-mail.

Not very clear. Pl try Laplace Transform