Patrick> ... does not apply.

Why not? Because you have to divide by zero in the textbook proposed solution?

It seems to me that your F(s) must be extendable to an analytic function in s, or the complex conjugate of an analytic function (depending on whether λ = +i or -i), with branchpoints at ±1, for a solution to exist. Because the left hand side can be expressed as a Cauchy integral encircling a branch cut from -1 to +1, and such integrals define analytic functions with the mentioned properties.

Maybe that is what Issam lead you to discover? Best of luck!

Dear Patrick,

Direct integration of both sides and collecting terms we obtain

ʃ G(s)ds = ʃ F(s)ds - (λ/π) ln(t-1/t+1)

where boundaries of the integrals from -1 to +1.

now substitute in the initial equation:

G(s) = F(s)- (λ/π) { ʃ F(s)ds - (λ/π) ln(t-1/t+1)}/ (t-s).

Hope it is useful.

Best wishes

Right, I am understanding your method; it is easy to solve, thank you. I was very tired. But, you missed G somewhere, so that your solution is incomplete.

Dear Patrick,

This problem can be solved in so many ways .However, one of the ways is by using direct integration of both sides of the equation. Thereafter, collecting terms which results into the following summarily:

step 1:

ʃ G(s)ds = ʃ F(s)ds - (λ/π) ln(t-1/t+1)

boundaries of the integrals from (s=-1(initial/lower limit to s=+1(final/upper limit).

step 2: substitution of condition after integrating both sides:

G(s) = F(s)- (λ/π) { ʃ F(s)ds - (λ/π) ln(t-1/t+1)}/ (t-s).

You can check mathematical methods specifically for ODE and PDE for other methods.

Hope it helps

Best regards,

Lateef.

Dear Patrick,

You get the method.Could you please read the file attached.

@ Lateef: a copy of the first answer.

In fact no need to pde or any advanced tools.

Best regards

Dear Patrick,

should (t-s) stand inside or outside the integral? You wrote it outside and so is also in Issam´s solution.

To Viera:

(t-s) stands inside the integral.

To Issam: I have from your method the right solution; this is the merit of ResearchGate. Thank you all

The solution of this integral equation is involved in my current study of interface cracks; I should present a paper on this work soon.

@Patrick: Wake up!

Sorry, but your last post makes absolutely no sense. What value of t in Issam's solution will you use in your paper? Didn't you read Viera's comment in full?

The equation is indeed of the Carleman type, as you mention. More specifically of the Cauchy type subclass of such equations. If the integration had run over ⟨–∞,+∞⟩, it could have been solved by Fourier transform (since the integral is of convolution type). But as I vaguely seem to remember, it is also solvable with a finite integration range, due to the wonderful properties of analytic functions. Except that there may be problems for some values of the parameter λ, which may make the operator not fully invertible.

Note added: I googled Carleman type integral equations, which returns many possible references. One is a full book, Chapter Carleman Type Integral Equations

which very likely covers your problem in detail.

Issam's solution in the attached pdf file does not apply because (t-s) stands outside the integral in his derivation whilst I look for a solution with (t-s) inside the integral. Hence the exact solution is not given at present. But, I feel able to find the right solution based on Issam's approach. For the value λ2 = -1, the solution proposed by Carleman (and in any book on Carleman-type singular integral equations) does not apply.

Patrick> ... does not apply.

Why not? Because you have to divide by zero in the textbook proposed solution?

It seems to me that your F(s) must be extendable to an analytic function in s, or the complex conjugate of an analytic function (depending on whether λ = +i or -i), with branchpoints at ±1, for a solution to exist. Because the left hand side can be expressed as a Cauchy integral encircling a branch cut from -1 to +1, and such integrals define analytic functions with the mentioned properties.

Maybe that is what Issam lead you to discover? Best of luck!

Exact, I have to divide by zero;

F(s )=as + (b + cs)√1-s2 where a, b and c are constants (reals).

Dear Patrick,

Divide both side of the equation by the factor (s-t) and proceed the same as in the first pdf file.

Integrate both sides, determine ʃ G(s)/(s-t)ds in terms of ʃ F(s)/(s-t)ds,

( in the first solution we eliminate ʃ G(t)dt, but now we eliminate ʃ G(s)/(s-t)ds )

finally, an explicit formula for G(s) is obtained ( including s and t).

@ Olaussen > Good interpretation.

Best wishes

Patric> Exact, I have to divide by zero.

You are not explicit about whether λ = +i or -i. Depending on this sign (and a little re-interpretation of your equation) you should find G(s) = ±i F(s)/(1±1),

where 1-1 means that no solution exists, because G(s) + (λ/π) ʃ dt G(t)/(t-s) = 0 for a huge class of analytic functions G(s). Maybe that can be circumvented by assuming that G(s) to be the complex conjugate of an analytic function. So I think the solution work lies more in a careful formulation of the problem, rather than clever algebraic manipulations. The problem looks very much like an application of the https://en.wikipedia.org/wiki/Kramers–Kronig_relations (which in turn is just a straightforward use of the Cauchy integral formula for analytic functions). Provided the functions are analytic. That is not obvious by just looking at expressions defined on the real axis. Note that √1-s2 is imaginary for s2 > 1, but with a the sign which depends on how one moves around the branch-points, and whether the extended function is defined to be analytic or not.

@Issam : I have your solution that depends on a parameter t ranging between -1to 1. Let see forward to see this result and its possible use in the study of interface cracks

@Kare: Both values of λ (±i) enter our work on the equal scale. Please see our contribution in ResearchGate “Interface crack under mixed mode I+ II loading and internal shear stresses due to Poisson effect: theory and confrontation with experiments” and associated comment No. 3 for a better understanding of what is going on. I have to recover your results. One can see your competence in the field.

It is a real pleasure to see your commitment to science (to all)

This is a quick reply that might already be covered by other answers but it is my attempt to help. Given that the 1/(t-s) is inside the integral (as it must be because t is a dummy integration variable), the integral equation looks to me like the Fredholm equation of the second kind. The kernal has the form that is discussed in problem 16.2.1 on page 735 of the textbook by Arfken, Mathematical Methods for Physicists, second edition, Academic Press. Methods of solution are given there but maybe you already know about that and have a special problem regarding singularities. If so, then sorry for wasting your time.

@Kare : I realize that I just need the solution for λ =-i and you provide the value G(s) = i F(s) / 2; this seems meaningful. But, at the moment, I do not have your solution method.

The solution at hand using the procedure suggested by Issam depends on a parameter t =-1 to 1; a general solution is further derived by superposition over t; this is made possible because the singular integral equation is linear. The value of the solution is zero at s=±1. Unfortunately, this procedure does not exclude possible singularities (oscillatory or not) at s=±1 associated with crack dislocation distribution functions (please see our works on the interface cracks to see connections).

A general procedure dealing with singularities at s=±1 is wanted when λ2=-1 .

I am happy to say that I agree with Kare solution from which I am able to extract the crack dislocation distribution functions. Impressively, it is possible to show that no oscillatory singularity exists at the tip of the interface crack under load when the integral equation under consideration applies. This is true under the requirement that the stresses are continuous at the crossing of the interface.

Associated paper of this Q&A question is presented; see "Non-planar interface crack under general loading III. dislocation, crack-tip stress and crack extension force" in our contributions in ResearchGate. I would greatly appreciate any comments. Thank you very much in anticipation