Dear Sir,

In my point of view, for a linear system of differential equations there exists an analytical as well as Numerical solution. We could compare the proposed Numerical method's solution with the exact one and we could estimate the error percentage. For both the cases we can plot graph to compare ... But in case of nonlinear it is not possible to find analytical solution( except few cases), only way to find solution is Numerical...

I respect your answer, thank you.

Dear Sir,

What I have read and learned so far is that there are two types of solutions, usually for equations or system of equations. This is the analytical solution and the numerical solution for both types of equations and their systems. There is a numerical solution, but the analytical solution only for linear cases. The nonlinear and complicated cases are solved by numerical methods and to prove the convergence of these solutions. The exact solution could be found ..

Greetings

if we can't find exact solution in nonlinear equations, how we can find the convergence and errors?

thanks

It is possible to find the posteriori errors. Numerically we can give the curve of errors

Professor Salah Abuasad,

I would like to discuss about some questions.

Your question:

What is the important solution: exact solution or approximate solution or analytical solution? 1. is it enough to give analytical solution? 2. can we give approximate solution only?

My opinion:

I think it depends on the research's purpose. If we solve a real problem, in many cases, approximate solution might be enough as long as the solution can be produced within an acceptable computational time limit. If the research aims to contribute theoretically, then obtaining exact or approximate solutions might be acceptable.

Your question:

3. what is the useful for comparing between the approximate solution and exact solution?

My opinion:

In order to assess the quality of the approximate solution, it must be compared with other good solutions, e.g., the exact solution. In general, an approximate solution can be produced in a shorter time than producing an exact solution when solving difficult problems. (NP-hard ones) Even the approximate solution can be produced faster, its quality needs to be assessed well

I hope you find my answers useful.

Best regards,

Ivan K. Singgih

Thanks .nice opinion.

Hi

Selection between the exact and approximate, is a matter of existence. If there exist exact solution then that should be selected. For natural phenomena there is no exact solution, and all solutions are approximate, with a degree of accuracy depending on the understanding of the phenomenon, and the previous efforts for its solution. My research team, developed a solution for all natural phenomenon, which is the most simple, and the most accurate solution in the literature. Look for the State Based Philosophy, and the Persian curve to in the literature and in the researchgate.

Engineering is the art of being approximately right rather than exactly wrong!!!

(Prof. Rod Smith)

Hello Mr. Ghaassempouri and all

Nice question, it seems to me that if convergence of the approximate solutions towards exact solutions can be guaranteed, approximate solutions are very suitable and mostly the most practical solution, especially for engineering applications is approximate solutions. By convergence I mean the capability to decrease the approximation and the errors as we want, up to the limits of round-off

Dear professor

The analytical solution is an exact solution to the model, but the model is an approximation to the physical reservoir.

The numerical solution is an approximate solution to the model, and the model is also an approximation to the physical reservoir.

As Abdolrasoul Ranjbaran is answered, it is true that the Selection between the exact and approximate is a matter of existence. If there exists an exact solution then that should be selected. Hence the analytical solution should exist. This Analytical solution may contain errors. An example of this the solution obtained by the variable coefficient for the initial value problem or direct integration. Thus Analytical solution is depending on the Analytical method. For natural phenomena, there is no exact solution, and all solutions are approximate, with a degree of accuracy depending on the understanding of the phenomenon, and the previous efforts for its solution. My research team developed a solution for all-natural phenomena, which is the most simple, and the most accurate solution for that problems.

Kedir Aliyi Koroche can you please explain this sentence again: My research team developed a solution for all-natural phenomena, which is the most simple, and the most accurate solution for that problems.

dear Dr.salah Abuasad, as I explain to you, the solution may exist as a form of an exact solution or an approximate solution for all-natural phenomena . But the problem is that how we can select those existing solutions is very simple to obtain by using either analytical or Numerical Methods and a more accurate solution. I mean that it depends on the method that we use to find this solution, thank you!!

Kedir Aliyi Koroche

ok clear

thanks

ok, Thanks@

It is significant that many equations of physics, chemistry, and biology contain empirical parameters or empirical functions. Exact solutions allow researchers to design and run experiments, by creating appropriate natural (initial and boundary) conditions, to determine these parameters or functions.

The solution which is more near to the reality is the most important. The method of solution, usually, is not important.

Classification of methods of solution is conducted because the way to obtain correct solution and the degree of approximation is not known a priori.

The exception is where the introduction of a specific method is on the table.