Dear Prof.Dr.Jafar,

In : Wazwaz A. M., 2011, Linear and Nonlinear Integral Equations Methods and Applications. Springer-Verlag, Berlin, the author stated:

The Adomian decomposition method consists of decomposing the unknown function of any equation into a sum of an infinite number of components. It was formally shown by many researchers that if an exact solution exists for the problem, then the obtained series converges very rapidly to that solution. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain.

Therefore in my opinion, when ADM gives the exact solution (analytic solution) we can say it analytic method. On the other hand, for complicated problems or problems which does not have an exact solution we calculate the error which becomes approximate or numerical method. By the way, sometimes they named the ADM, HAM, HPM, VIM…etc as ‘A semi-analytical iterative methods’.

Best regards,

Dr.Majeed

Dear Prof.Dr.Jafar,

In : Wazwaz A. M., 2011, Linear and Nonlinear Integral Equations Methods and Applications. Springer-Verlag, Berlin, the author stated:

The Adomian decomposition method consists of decomposing the unknown function of any equation into a sum of an infinite number of components. It was formally shown by many researchers that if an exact solution exists for the problem, then the obtained series converges very rapidly to that solution. However, for concrete problems, where a closed form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. The more components we use the higher accuracy we obtain.

Therefore in my opinion, when ADM gives the exact solution (analytic solution) we can say it analytic method. On the other hand, for complicated problems or problems which does not have an exact solution we calculate the error which becomes approximate or numerical method. By the way, sometimes they named the ADM, HAM, HPM, VIM…etc as ‘A semi-analytical iterative methods’.

Best regards,

Dr.Majeed

Thanks for your reference.

As I understand you agree with Prof. Wazwaz.

Interesting Question. Following. Best regards.

Thanks for considering this question.

I know this methods as Numerical!

What do you think.

Its nice reply by Dr majeed.

I consider ADM and HAM s numerical methods....but Dr Majeed reply is more affective ....

Good luck

M N Ali

In my opinion, it is preferred to be called "numeric-analytic methods" as it could be utilized for both purposes.

Regards- Mohammed

Dear Prof. Jafar Biazar

I think we call it analytical methods are more appropriate due to the solution for the problems using these method give you in first glance a sequence of analytical functions rather than numerical.

Best regards

Mohammed

I think in my opinion that we must use the abbreviation which was used for the first time!!!

we can't always define the solution as numerical solution or analytical solution depending in exact solution or approximate solution. we need more effective rules.

Dear Prof. Jafar Biazar,

In my point of view, theses are called analytical methods as i read, many researchers used to find the analytical solution.

Regards

Muhammad Nadeem

Dear Prof. Jafar Biazar

I agree with Prof. Dr. Majeed Ahmed Weli

Wonderful and Excellent Question.  

Best regards.

Dear Prof. Jafar

I totally concur with Dr. Majeed Ahmed Weli.

These methods are referred to as semi-analytic methods as they involved no descritization. In my opinion, approximate methods are generally divided into two viz: Semi-analytic methods (ADM, HPM , HAM, VIM ,etc) and Numerical methods (RK, Adams, Nystrom, etc).

Thank you and kind regards

Dear Prof. Biazar,

I think the numerical or analytical solutions depends on the fact that how problems were been organised (linear and nonlinear, definition of derivatives -partial, fractional or It's and etc) the vector space of solution (Weak and strong solutions)

In most of the problems researchers should attempt to find analytical solutions, clearly numerical solution for which is just parallel research.

For those problem which have not analytical solutions yet, the numerical solutions would be a great discovery to illuminate the way to find the proper weak and strong solutions.

Sincerely,

Dear Dr, Safavi,

I would like to thank you for your contribution.

Those are semi analytic methods

We say analytic method when we get closed form solution

Dear prof. Biazar.

It is a good question to propose. I think if you evaluating the solution at some points for a specific interval " mesh or grid points" then it is a numerical approach as in collocation methods. But working on intervals where the solution is convergent, we call it analytical.

Regards

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Hi. I have done all of my anlysis with Abaqus software in which is very precious especially when someone want to simulate metal. Abaqus consider all of the boundry condition very well

I think these solutions are analytical solutions. Although it is an iterative method. This method is similar to Picard's iterative method.

I think it is analytical methods because through these methods we can get the approximate solutions in an analytical form.

classifying the methods to two types as analytical or numerical methods, or some times semi-analytical methods is very difficult in most cases, it is still confusing point, since:

1. we define the analytical methods when it give us exact solution, and numerical methods if it give us approximate solutions, and the semi-analytical numerical methods if is it give us exact solutions or/and approximate solutions depending on the type of equations and its linearity. depending in this methods, almost all method are in third type, semi, since it give us exact in some cases and approximate solutions in other cases.

2. in some cases we can find the exact solutions, but we use approximate solutions for many case: to decrease the cost of finding exact, it is very difficult to find exact, the approximate solutions give us a very good solutions when we compare the approximate of proposed method with the exact solution of previous methods, that is the absolute or relative errors are very good.

in my opinion, I think it is very difficult, or sometimes (useless) to classify methods as numerical or analytical.

thanks

i think both are same analytical methods

I think this is an important discussion. I am of the opinion that ADM, HPM, etc are analytic(al) methods, though approximate. Therefore they can be rightly referred to as approximate analytic methods. This means that analytic methods can be exact or approximate, and can be based on series solutions or closed-form solutions.

In my view, the distinction between analytic and numerical methods is that numerical methods are discrete techniques that derive their solution on a point by point basis while the analytic schemes are based on continuous functions that apply to a range or all of the solution domain.

@ Akuro Big-Alabo

the distinction between analytic and numerical methods is that numerical methods are discrete techniques that derive their solution on a point by point basis while the analytic schemes are based on continuous functions that apply to a range or all of the solution domain.

can you give examples for your opinion?

Dear Prof. Salah Abuasad,

What do you think about Interpolation, or Splines for approximating a function.

Approximation Techniques are Numerical methods, and ADM, HPM,... find an a approximation to Taylor expansion of the solution, and since we can recognize the function, Many researchers call these approaches Analytic.

I think analytic approaches lead to exact solution, while Numerical lead to pint to point solutions or and approximate to analytic solutions.

so, simply,

analytical solution describe exact solution. and approximate solution describe numerical solution??

Jafar Biazar

The limitations of analytic methods in practical applications have led scientists and engineers to evolve numerical methods, we know that exact methods often fail in finding root of transcendental equations or in solving non-linear equations. There are many more such situations where analytical methods are unable to produce desirable results. Even if analytical solutions are available, these are not amenable to direct numerical interpretation.

Dear Dr. Muhannad Shallal

Thanks for your participation.

Dear professor, in my humble opinion, we just use it instead of symbolic computation. The better words I believe would be symbolic solutions and numerical solutions. Symbolic solutions need not to be exact solutions. Especially the series solutions are mostly approximate solutions.

Dear Dr. Ahmed,

Thanks for your participation

Regards,

Jafare.

I think these methods are semi analytic method not analytical methods ,since these methods not always gives accurate(exact) solutions.

Semi analytic methods will be an appropriate word because of the series solution which is always an approximation, after doing traction. Also in HAM we use the initial guess, which always leads to an approximation.

According to me, both of them have some advantages and drawbacks and can be used in several fields. It is clear that Analytical solution is much more exact and generally has been used by researchers for calculating theoretical work and result. I believe that this method can be so useful for testing some theories and numerical methods for spacial cases for finding the best results. For example, some times researchers have a complex system and the only way for calculating is numerical method. However, if we want to find the best numerical method, we can test our numerical method with use of analytical method in some simple case in order to find the best way in complex case.

Dear Dr. Sobhan Kazempour

I would like to thank you for considering my question.

Regards,

Jafar.

Math. teacher.

I think these methods are semi analytic method not analytical methods ,since these methods not always give accurate(exact) solutions sometimes give as undefined series solution(not as series of cos, tanh,...). So, we must solve it numerically by any software.

Regards

Dear Dr. H. Oda Al-Humedi,

I would like to thank you for considering the question.

Jafar.

These are of course analytic methods, if one can prove that the complete series is convergent mathematically. However, since such a proof is very much tedious and frequently impractical, due to the complicated nature of the series, only finite number of terms are evaluated numerically, therefore, we call them semi-analytical.

Semi Analytic

Najmuddin Ahmad why some researchers divide to numerical or analytical or semi??

can you write the term fully

In my opinion these are semi analytical methods as they do not always give exact solution. In various complicated cases they provide approximate solutions.

Analytical solutions are in terms of an infinite series that converges to the exact one. The n-th partial sum of this series is considered to be an approximate.

If you assume the same mathematical problem, the difference in the numerical method is in the fact that you have a finite dimensional vector as solution that, depending on the accuracy and computational steps of the method, approximate the analytical solution. Convergence of the numerical method ensures you asymptotically get the same analytical solution.

More in a general framework, a difference-key is that we introduce suitable simplifications in the original mathematical problem to get a closed solution while we work numerically on more complex set of equations that have no closed solution. Therefore, we should consider an exact solution of a simplified problem versus a numerical (apporximate) solution of a more complete set of equations.

Analytical solutions of mathematical problems are solutions that are exact and in closed or quadrature form. Sometimes, these solutions may not be usable as they are for practical purposes and approximating these values to a predetermined degree or error of precision/approximation is used to the sought solution and works fine. The latter process of crafting working solutions is what we call numerical or approximate methods.

interested

Let me add some further point: if you get the first terms of a series that is supposed or can be shown to converge to the true solution, one can use convergence acceleration or extrapolation methods on the partial sums of the series to get better approximations to the true solution. Such methods can work on numerically or analytically given partial sums of the series. For instance, in the case of power series, one can use Pade or Levin-type extrapolation methods to to obtain rational approximations that can allow you to estimate pole singularities of the true solutution. If you need more information on such methods, you can find references in my list of publications.