it really depends on the form of nonlinear term (or terms). would be nice if you add a prototype of the mathematical PDE/ODE system here?

as you may know a problem of solving a nonlinear PDE will reduces to solving a system of nonlinear allegoric equations. Thus in A SENCE nonlinear PDE is equivalent to a nonlinear algebraic system. The most direct and elegant method is newtons method where you need to build so called Jacobian matrix. In case the simple Newton's method is not good (highly nonlinear), one may go to so called, Two-point newton's method and at finally another class of methods call Halley's method or super-halley's method can be helpful. To my experience, Halley's method can be hard to understand and implement and not so rewarding. In practical cases, it is worth more to work on finding a better initial guess for Newton's method than changing the Netwon's method to Halley's method. Carefully read the following links:

https://en.wikipedia.org/wiki/Newton%27s_method

https://en.wikipedia.org/wiki/Halley%27s_method

I have solved many high order nonlinear simultaneous equations using several techniques. Can these equations be expressed in some transferable way? (formulas, an Excel spreadsheet, etc.) or are you trying to control some complex model of something so that running the model must be inside the calculation loop? I've done that many times too and published a paper on it.

Nonlinear system of algebraic equations in n-variables are numerically solved using:

1) Newton's method

2) Chebyshev's method

3) Halley's method

4) Super Halley's Method

There are other derivatives of the above methods.

Thanks!

Hi Mr Farzad Abdollahi .

I agree with:

Dr Vahid Mosallanejad

Dr Dudley J Benton

Dr Tekle Gemechu

Thank you.

Dear

Aref Wazwaz

thanks!

What method you choose also depends on whether or not you have the partial derivatives of the residual with respect to the independent variables in explicit form. If you do, then you can use several methods, including Newton's. The Davidon-Fletcher-Powell method is also a good option (see article on Wikipedia for this and associated methods). These are available in MatLab. If you don't have the partial derivatives or they are too burdensome to calculate, you can use something else, like Broyden's method, which doesn't require them. An example of when you don't have the partial derivatives explicitly would be tuning a model. Let's say you have a model of the atmosphere or a lake or a factory or a fast food joint and you want to find the optimal "tuning" factors to best represent observed behavior. It may take hours to run a single simulation, comparing model results to measured values. You could simulate Hurricane Katrina inside Broyden's method and let it run for a week or a month or a year. I have done this with several large lakes to tune empirical models for biological oxygen demand. There are several animated examples here http://dudleybenton.altervista.org/projects/WaterQuality/index.html

Hello and thank you very much Dudley J Benton , Tekle Gemechu , Aref Wazwaz , Vahid Mosallanejad

honestly, I solved my equations numerically with the help of a combination of maple software commands like (seq, fsolve complex, ... and plots package).

Of course, I will use your tips to solve the problem at the next level but right now I'm satisfied because the results are satisfactory. It should also be noted that Newton-Raphson method did not work for our system of equations because although the algorithm I had written would solve ordinary equations system very quickly and converge after several steps even with the initial values far from the root(with GNU Octave ). I was going to perform the broyden's method, but as I say before my results were obtained with Maple.

You are most welcome dear Tekle Gemechu . Wish you the best always.

You are most welcome dear Farzad Abdollahi

Wish you the best always