All these methods are semi analytical in nature give generally a solution in a series form which can be written in closed form if you know what this series represents. Any way they are good to solve nonlinear problems. Their effectiveness can be measured by knowing the computational efforts put into and quality of solutions obtained. A method that can give a solution with ease is a good method, numerical or otherwise. Some time you may combine the two.

Thanks Prof.R. C. Mittal

Dear

You may study the difference between two successive approximate solutions u_(k+1) -u_k.

Also, you may check the ratio of the coefficients for two successive terms in your analytic solution.

You may compare it with exact solution in case it is available.

Regards

Thanks Prof.R. C. Mittal for you

One can use such methods to judge the convergence speed by some analytical methods, This is often more general than just at looking at numerical results that usually can be obtained only for certain test cases.

You may use the ratio test |a_k+1 /a_k| < 1

These methods are approximate numerical methods that give the solution in the series form that may converge to the exact solution and may not converge, giving an approximate solution

Here we can not generalize a particular method to all nonlinear equations for each method of effectiveness in certain equations and not all

A certain method may solve a certain type of equations and the other fails to solve the same equations and vice versa if the equations are changed

I agree with Prof.Elzaki,

These methods find first terms of the Taylor expansion of the solution.

We recognize the solution. If we don't recognize, the results would be an approximate to the solution in analytic form.

So these methods do not lead to the solution, themselves.

Thanks Prof. Biazar

I recommend the answer of Mr Elzaki Tarig

I agree with Prof. Dr. Elzaki,

The answer by Prof.Dr.Elzaki is good. However, to check the effectiveness of the approximate methods, you have two cases:

a) If the exact solution exist then simply calculate the absolute error ABS[exact-approximate], less error mean better accuracy.

b)If the exact solution is unknown(which is in many nonlinear and practical problems) you have to evaluate the maximum error remiander.

Thanks

In the usual way-by checking how accurately they represent the ``invariant'' properties of the equations being solved. That's what, unfortunately, isn't, usually, mentioned.

The error residue isn't the most effective way to describe global properties.

Incidentally: the term ``semi-analytic'' doesn't mean anything; nor does the term ``numerical''. Both can be programmed for treatment by computers and both can be analyzed regarding how well they represent the properties of the equations.

Thank you all

Thanks to those agreed with me.

Thanks professor Biazar

Dear all, Since 20 years, I have worked in that field. You come to my working area.

Please correct my answers if I mistook. I’ll respond to all of you in points.

1st point: Mohsin Abdullah Al-Shammari , R. C. Mittal

What I know about solutions. The solution is analytic or not analytic. There is a definition for analytic. If you have a definition for semianlaytic, please give it to me.

The difintion of analytic is “In mathematics, an analytic function is a function that is locally given by a convergent power series….”

2nd point: What I know is that the Taylor series is proofed as a convergent series for any function as the derivatives exist. Any other series must be proofed….OK?

All the method that stated before and others give formulas to obtain series solution and it is success if the obtained series is Taylor series and the solution is very good (in limited region because you can’t obtain terms up to infinity).

But in some cases, the formulas can’t work with all problems and don’t give the Taylor series and the solution is not accepted.

For more details see:

Improved Adomian decomposition method

Article Improved Adomian decomposition method

Conference Paper SOLVING NON-LINEAR KLIEN-GORDON-TYPE EQUATION USING IMPROVED...

Modified variational iteration method

Article Modified variational iteration method (nonlinear homogeneous...

Article Modified variational iteration method (non-homogeneous initi...

Marwan Alquran, Mohamed R. Ali

You can compare with the exact solution if you introduce a new method and need to test it and if you succeed it is not a proof that the method will work with all the other methods. So if you don’t have an exact solution what will you do?

Another point, comparing two successive term is not easy as you mentioned.

Elzaki Tarig , Herbert H H Homeier , Jafar Biazar , Majeed Ahmed Weli , Stam Nicolis .

I agree with you 100 percent. I tried to solve all the problems in the previous methods like (convergence and limited region of solution and upper bound of the errors)

And introduced a new method and call it “Piecewise Analytic Method PAM”

I have just an accepted paper and others will be published soon

Preprint Piecewise Analytic Method (PAM) is a New Step in the Evoluti...

I compared PAM with Runge-kutta method

Preprint Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)

I opened a discussion to learn from you and correct my mistakes

https://www.researchgate.net/post/Is_there_a_method_can_compete_Runge_Kutta_method

https://www.researchgate.net/post/What_do_you_know_about_Piecewise_Analytic_Method_PAM-Published_January_2012_by_T_Abassy

Dear Dr. Tamer Ahmed Abassy ,

I would like to thank you for your informative message.

Please send me a copy of your accepted paper, when it is published

Regards,

Jafar.

Dear Prof. Jafar Biazar ,

Thank you.

The paper is published and available for downloading on the URL:

Research Piecewise Analytic Method (PAM) is a New Step in the Evoluti...

Best regards,

Tamer Abassy

Recently I had a paper to evaluate the homotopy analysis method based on the stochastic arithmetic and the CADNA library to find the optimal step of HAM, optimal approximation, optimal error and optimal h. Please find papers in my profile. Also I am working to evaluate the homotpy perturbation method now. See these papers:

1- Finding optimal convergence control parameter in the homotopy analysis method to solve integral equations based on the stochastic arithmetic

2- Valid implementation of the Sinc-collocation method to solve linear integral equations by the CADNA library

3- Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method

thanks docteur Samad

Dear friend

I also haven't work these methods.

Thank You Mr. Samad