Partial integro – differential equation have played, expecially in the past,  an important role and a great interest in the  mathematics comunity.  

In litterature there are a huge number of methods for finding solutions of this type of equations, both numerically or analytically. But, in general, each method works well only for a specific class of problems, often solving a single example.

My sincere advice is, before diving into the  vast litterature, to understand what kind of problem are you having to attach and, only after this phase, look for a suitable method.

Take a look at those papers:

1) "The Numerical Solution of Coupled Integro-Differential

     Equations" by Pennell and L. M. Delves, a classical (60's)!

2) "EXISTENCE RESULTS FOR SOME NONLINEAR FUNCTIONAL-INTEGRAL

     EQUATIONS IN BANACH ALGEBRA WITH APPLICATIONS"  by LAKSHMI       NARAYAN MISHRA, H. M. SRIVASTAVA AND MAUSUMI SEN,

the proposed method works for various integral and functional equations

considered in nonlinear analysis and in applications,

or, if you can find it, this book:

O'Regan,D.,Meehan .M, Existence Theory for Nonlinear Integral and integro-differential Equations, Kluwer Acad.Pulbel. Dordrecht, 1998.

I hope I was helpful,

Fabrizio

For partial differential equation, I suggest works of V. I. Arnold, and Jerrold E. Marsden in Hydrodynamics. But generally existence of solution for PDE is a hard problem and one of open problem in mathematics (Navier-Stokes equation) is in this caAdd your answertegory. Linear cases are trivial and can be solved analytically. Nonlinear cases not even solvable, but existence problem is hard to prove.

Partial integro – differential equation have played, expecially in the past,  an important role and a great interest in the  mathematics comunity.  

In litterature there are a huge number of methods for finding solutions of this type of equations, both numerically or analytically. But, in general, each method works well only for a specific class of problems, often solving a single example.

My sincere advice is, before diving into the  vast litterature, to understand what kind of problem are you having to attach and, only after this phase, look for a suitable method.

Take a look at those papers:

1) "The Numerical Solution of Coupled Integro-Differential

     Equations" by Pennell and L. M. Delves, a classical (60's)!

2) "EXISTENCE RESULTS FOR SOME NONLINEAR FUNCTIONAL-INTEGRAL

     EQUATIONS IN BANACH ALGEBRA WITH APPLICATIONS"  by LAKSHMI       NARAYAN MISHRA, H. M. SRIVASTAVA AND MAUSUMI SEN,

the proposed method works for various integral and functional equations

considered in nonlinear analysis and in applications,

or, if you can find it, this book:

O'Regan,D.,Meehan .M, Existence Theory for Nonlinear Integral and integro-differential Equations, Kluwer Acad.Pulbel. Dordrecht, 1998.

I hope I was helpful,

Fabrizio

An excellent book is:

Partial Differential Equations by Lawrence C. Evans, Graduate Studies in Mathematics, Volume 19, American Mathematical Society, Rhode Island (1998). This is a very readable textbook, that is also highly estimated by some of my colleagues in mathematics. Warning: Results on the existence of global (not only local) solutions are meagre.

Dear  Professor Erik Baxter,

To the best of my knowledge,the following two books may be helpful to solve your problem.

1.G. S. Ladde; V. Lakshmikantham and A. S. Vatsala;Monotone Iterative Techniques for Nonlinear Differential Equations,Boston, Pitman, 1985.

2.C. V. Pao; Nonlinear Parabolic and Elliptic Equations, New York, Plenum Press, 1992.

Existence theory of  Nonlinear Partial Differential Equations is well studied by developing and applying the constructive method known as Monotone iterative Technique .

In a short pan of time I can explain this technique which is simple and elegant. I taught M.Sc. course (two semester) at  Dr.BAM Univesity,Aurangabad,India on the said topic.

Best luck

A helpful book regarding nonlinear dynamics that I have used is:

Nonlinear Dynamics and Chaos by Steven H. Strogatz; Westview Press (a Perseus Books Publishing, LLC)--1994.  ISBN-13 978-0-7382-0453-6  ISBN-10 0-7382-0453-6.

As various people have said, the existence question is really a set of questions quite different for different examples/problems.    Now the most frequently seen approaches start out by seeking some form of weak solution as a limit in some large space with the equation interpreted in a quite generalized sense (weak solution, distributional solution, variational solution, viscosity solution,...) and then look for regularity results to hope these are solutions in something closer to a classical sense.   The original existence results are typically obtained by the use of abstract results in which the operators relate to variational forms or degree theory.   The hardest part is finding the right setting to show some form of compactness (weak topology, Aubin-Lions Theorem, etc.).   It is often at this point that ideas become very different for different problems.  Note that the most frequent approaches often start with the modification of "similar" known results.  It would possibly help if you would be more specific about your problem. Of the books mentioned, I am most familiar with Evans' book and will recommend that.

Dear Professor Erik Baxter,

With respect to integro-differential equation may be attached text book help you.

Best Regards

My guess is the equations are non-linear.  There are numerous general methods for solving linear equations - depends on the type, elliptic, hyperbolic or parabolic.  When non-linearatines creep in there are few "general theories" and the techniques depend on a laundry list of methods for index theory, fixed point theorems, etc.  Each equation will in general require its own method.  A good place to start is Michael Taylor's three volume series on Partial Differential Equations,

Vol I - Basic Theory, Vol 2 - Qualitative studies of linear equations and Vol 3 - Nonlinear Equations.

I know that a lot of work on proving the existence of solutions of PDE and non-linear systems emerging in fluid dynamics was done by Russian mathematician Olga Ladyzhenskaya; see https://en.wikipedia.org/wiki/Olga_Ladyzhenskaya for finding some of her works. I even was lucky to be a student of one of her courses more than 30 years ago.

She was also dealing with integro-differential equations and there are papers devoted to her, like this one: http://link.springer.com/chapter/10.1007%2F978-1-4615-0777-2_11#page-1 . But I agree that the topic is very difficult for a beginner and you better search for results for your particular equation type, or the class to which it belongs.

As an engineer, not a mathematician, my usual approach to the existence question is to produce an example solution. If I can't do that then I have a problem. Uniqueness theorems are much easier to prove than existence. If I can't produce an example solution and don't know of a theorem guaranteeing a solution, I don't have much confidence in existence.

I think an approach that is commune in finite element is the use of the weak formulation to prove the existence and uniqueness of the solution. one has to prove that a certain linear functional involved in the PDE's is elliptic which implies the existence and uniqueness. look into the work by Franco Bezzi. such as:" On the existence, uniqueness and approximationof saddle-point problems arising from

 lagrangian multipliers".

@Erik

Use Matlab for simple problems, hope you will find your way to

address the problems of yours.

Prof Rath

Dear Dr. Baxter,

I think any books on PDE, can help you,

Have you tried google search?

yes it can help

You can use the Glarkins method to prove a local presence which is a good way to prove local existence.