I played the mathematical physicist game for 25 years and then have studied how people learn and think about physics for another 25. My answer is both no and yes.

For the "no" part, let me tell you a story of a research project I did back in the '70s. This wound up requiring an integral of a complicated polynomial fraction. Having studied complex variables, I knew there was a way to calculate an analytic solution. (For the technically minded, this used Cauchy's theorem and converted the integral to the residues at the complex poles.) We were very pleased with this and checked our work carefully. When we then tried to calculate it by putting numbers into our analytic solution, we got a shock. The graphs we calculated wandered wildly all over the page! What we discovered is that the analytic solution expressed the result as a difference of some VERY large numbers. If we only kept 8 decimal places in our calculation we got nonsense. To see a reasonably accurate answer we had to keep 16! But looking at the integrand, it was very smooth. We could get an answer to 10% with a very crude and easy to calculate approximation, and we could get an answer to 5 decimals much more quickly and easily using a standard numerical method to calculate the integral than with our "analytic" solution.

For my "yes" answer, I'll argue that in exploring how you might get an analytic solution, you'll get tremendous insight into the solution and how it behaves, whether you in the end have to wind up using a numerical approach to calculation or not. To find your analytic solution you should identify how the parameters of the problem cluster, how they create scales in space and time (or whatever the variables are) that are natural to the problem.*

You should look for any danger points -- places where something goes to 0 or gets very large -- places where you expect something to settle down to a simple form or go away where perhaps it doesn't. The insights you get from exploring analytic possibilities, if they don't give you an analytic answer, help you a lot in finding a numerical method that works better.

An example of this was using the differential equations of quantum mechanics to calculate the binding energy of the helium atom. The electric potential energy is infinite when the two electrons are on top of each other. This makes the wave function behave weirdly at that point. If you don't put into your numerical calculation that tiny piece of the wavefunction (which you can easily determine analytically), your numerics will spend all its time trying to get that piece right.

So my advice is "Think analytically, but don't assume analytic is always the best way to calculate. Stay light on your feet and use the mix that is right for the problem you have."

* http://umdberg.pbworks.com/w/page/68855405/Dimensionless%20equations%20%282013%29

Analytical solutions either total or partial are always preferable and an effort should be made in that direction in the phases of modeling and formulation. However when such solutions are not possible without losing the essence of the problem or the practical applicability, numerical solutions can be applied currently with a high level of accuracy and precision using more or less conventional numerical methods or association of methods. From an Engineering point of view, the focus should be made mainly on the best way to solve the problem in hand. For example, in mathematical programming, an exponential time algorithm is known to work really badly when the dimension of a system grows, but may be very effective and even a very useful tool for systems of little dimension.

Sir, as per my knowledge goes, analytic solution is always better than any solution for its versatility and it gives result much faster than other solution. Sometime, we need approximation, but still the solution is valid for a wide range. Curve fitted solutions are also faster but the solutions are mainly valid for limited range.

mathematical model is enough for a research. analytic solution is another research. one good example is also a research in mathematics. From original idea to commercialized product there is a long chain, every node of the chain is a research and development.

My thinking is that if you can find an analytical solution, it is good to do so. If for no other reason, it's a good way to confirm whether your math model is on the right track.

The problem is, in my experience, analytical solutions are frequently limited to just a simple set of conditions. So they are not truly more accurate than all the solutions a math model can provide. Analytical solutions to complex problems will easily become intractable, which is why the typical approach to coming up with a nice, clean analytical solution, is to use simplifying assumptions.

I agree to the idea of Albert, 

A mathematical model correspond to a number of analytical solutions. Model is at an abstract level, while analytical solutions is at a concrete level, like an example. Of course, we can use example or an analytical solutions to verify the model. 

From the above discussion, it is found that Mathematical model is simple enough to solve a particular problem. If analytical solution is available, then it is preferable. I agree. With my limited knowledge, I want to add one thing here as a researcher. After solving one problem using mathematical model, we have to try to find the solution analytically. Otherwise, unsolved problem will remain unsolved..

A mathematical model must be control with analytical models because this control accept the mathematical is find in the real  action.

A research problem formulated as a mathematical model can be of two types (from my experience): Optimization or Analytical. So whether Analytical or Optimization, its tricky and usually we find a problem discussed in past that is isomorphic to our research problem. This is done before a mathematical model is developed.

Disclaimer: These are my view specifically. Please discuss this with other members and ur supervisors (if any) before taking a peep into my viewpoint.

if analytic solutions are available, it certainly does make sense to find them. Even if such solutions for some reason turn out to be of no direct relevance for the intended applications, they can be employed e.g. as benchmarks for testing numerical solution methods.

Sorry for the delayed response. I feel the question itself is confusing. A mathematical model is in itself an approximation of a physical event, which can be mechanical, chemical, biological etc. An analytical solution will be possible depending on many situations. First, are all the variables involved continuous in nature. If the variables are discrete, result will again be approximate depending on the extent of discrete values available and the intervals in between. Unless too accurate answers are demanded, in today's perspective, attempting analytical solution may not be advisable. As such attempts may also be very difficult. However, if analytical solutions are possible, then that's definitely more desirable.  

An analytical solution is preferable mainly for the greater “guarantees” it gives, e.g. with respect to the problem of the convergence of a numerical solution. For much time Mathematicians have rejected solutions acquired by means of numerical methods, but some of this methods have been accepted because assessed as able to give convergent solutions.

On the other hand, many important problems of applied mathematics couldn't have found a solution other than that numerical. And there are "historical" problems for which a general analytical solution, that be not limited to some special case, does not still exist; one for all, the "three-body problem" in Mechanics

We should go for Analytical Solution as well. It gives a better support to our findings and conclusions

I think there is not any doubt, an analytical solution is always the best option.

Obviously, you have to keep in mind a couple of things:

a) Maybe, your model has not an analitycal solution.

b) Maybe, it has such a solution but you are not able to find it.

In both cases you could think that time used trying to find it, is a lost time and therefore you could used it in other more productive things. But this way of thinking is wrong, this time is used in understand your model more precisely and even to refine it. Who knows if some tuning of the model keeps the essentials and converts it in "analytically solvable".

In the opposite, if you are able to find such solution, you have all the variants and the influence of all the parameters that, probably, has your model. That is unvaluable because a numerical solution is only the solution for a given selection of parameters values. And therefore if you change any of them you must to solve numerically again. It is more time-consuming to explore in parameter space that the time you employ to find an analytical solution (although both can tend to infinity)

I played the mathematical physicist game for 25 years and then have studied how people learn and think about physics for another 25. My answer is both no and yes.

For the "no" part, let me tell you a story of a research project I did back in the '70s. This wound up requiring an integral of a complicated polynomial fraction. Having studied complex variables, I knew there was a way to calculate an analytic solution. (For the technically minded, this used Cauchy's theorem and converted the integral to the residues at the complex poles.) We were very pleased with this and checked our work carefully. When we then tried to calculate it by putting numbers into our analytic solution, we got a shock. The graphs we calculated wandered wildly all over the page! What we discovered is that the analytic solution expressed the result as a difference of some VERY large numbers. If we only kept 8 decimal places in our calculation we got nonsense. To see a reasonably accurate answer we had to keep 16! But looking at the integrand, it was very smooth. We could get an answer to 10% with a very crude and easy to calculate approximation, and we could get an answer to 5 decimals much more quickly and easily using a standard numerical method to calculate the integral than with our "analytic" solution.

For my "yes" answer, I'll argue that in exploring how you might get an analytic solution, you'll get tremendous insight into the solution and how it behaves, whether you in the end have to wind up using a numerical approach to calculation or not. To find your analytic solution you should identify how the parameters of the problem cluster, how they create scales in space and time (or whatever the variables are) that are natural to the problem.*

You should look for any danger points -- places where something goes to 0 or gets very large -- places where you expect something to settle down to a simple form or go away where perhaps it doesn't. The insights you get from exploring analytic possibilities, if they don't give you an analytic answer, help you a lot in finding a numerical method that works better.

An example of this was using the differential equations of quantum mechanics to calculate the binding energy of the helium atom. The electric potential energy is infinite when the two electrons are on top of each other. This makes the wave function behave weirdly at that point. If you don't put into your numerical calculation that tiny piece of the wavefunction (which you can easily determine analytically), your numerics will spend all its time trying to get that piece right.

So my advice is "Think analytically, but don't assume analytic is always the best way to calculate. Stay light on your feet and use the mix that is right for the problem you have."

* http://umdberg.pbworks.com/w/page/68855405/Dimensionless%20equations%20%282013%29

It depends upon your aims. Bolsmann did not find anslitical solution of his equation, but proved H-theorem, and so introdused entropy.

It is always good to find analytical solution. However, this is not simple to achieve. We rely on the expertise to apply advanced techniques mixed with optimization routine to solve the problem in particular constraints. Unfortunately, we loose the general case in most studied problems.

Thank you very much for all the valuable comments. I believe there are two types of scientific research one is an open ended research that aims at the understanding of physical phenomena, deriving new configurations and applications etc.  The second one is goal oriented and needs to be completed possibly “yesterday”.  Second one is dependent on the first. More precisely we denote the second type by “research and development”.  For the first type of research understanding is more important than implementation and therefore we cannot afford to lose any information by using numerical techniques unless it is unavoidable..

@ Prof Bagchi, It appears that the question is confusing. It is therefore a responsibility on my part to justify the same. Mathematical model definitely hides many aspects of the actual system. It is desirable also to highlight the required characteristics of the problem while introducing abstraction for the other parts of the system. However, one should be cautious, while developing the mathematical model, such that abstraction does not clutter or delete the required elements of the problem or change its characteristics.  Discrete nature of the problem does not create any difficulty to get analytical solution. A very simple example for this is digital signal processing.  Many other examples do exist.  Moreover I believe accuracy is not the only requirement in support of analytical solution, it is the insight that is important.

Dear  Edward,

I think your advice is the best

 "Think analytically, but don't assume analytic is always the best way to calculate. Stay light on your feet and use the mix that is right for the problem you have."

Regards,

Anup

This is a good and useful discussion.

The only thing I would add to the many excellent answers is that numerical models usually involve writing programs, and programs have bugs, and even when not, they often involve parameters that must be chosen appropriately. One way to find the bugs and test the adequacy of the chosen parameters is to test the model in those (often very limited) regimes where analytic solutions are possible and even simple.

I often tell students to assume that their programs have bugs until proven otherwise. The same goes for complicated analytical calculations where algebraic errors are common. It is always best to confirm a result in at least two different ways.

When I coauthor a paper, I usually insist that at least one other coauthor do an independent check of my calculation, whether it is numerical or analytic. I usually only review papers where I am willing to check the results myself, since more often than not, I find errors in the calculations. This is especially annoying when a paper has many coauthors.

It depends on the final objective. If you want to design a control strategy the analytical solution will be important. 

Verification of a program by checking against a reference solution is extremely difficult. In the slides gcd1 through gcd 3 I have shown that an algorithm claimed to be correct for gcd computation passes through four successive checks as depicted in gcd1 and gcd2. However it fails in gcd 3. Moreover analytical check of the algorithm can also refute its correctness claim. It is true that I have tailored this example and it is definitely an exaggerated representation but the underlying truth may not be impossible.

Correctness is the major issue against numerical solution (By “correctness” I do not mean accuracy).  Mathematical model of a distributed digital system may be obtained by specifying the constituent processes and their interaction scheme in a formal specification language.  A researcher may propose a new architecture and claim some advantage over the similar existing systems.  Before any such claims one needs to ensure the safety and liveness property of the proposed system.  For this purpose numerical solution technique will not give us any help. To ensure deadlock freedom, for example, exhaustive numerical simulation of the system for all possible states may be computationally prohibitive if not impossible. One has to rely upon some logical tools for its analytical proof. Inference drawn on the result obtained from a considerably large number of simulations will never ensure correct conclusion.

This is a great question.    In addition to the excellent observations that have already been made, there is a bit more to add.

@António Manuel Abreu Freire Diogo: Analytical solutions either total or partial are always preferable and an effort should be made in that direction in the phases of modeling and formulation.

I agree,   But then there is the underlying issue of the pervasiveness of various forms of analytical solutions that befit particular mathematical models of research problems.

Keeping as close as possible to the first part of the question "mathematical model for a research problem", perhaps we should consider alternatives to "analytical solution" in the second part of the question.   Here are a couple of suggestions;

Proof: if a research problem is either pure mathematics or theoretical physics, then it would be appropriate to couple a "mathematical model" with a proof or derivation of a solution.   Helpful examples of derivations are given in

https://www.researchgate.net/profile/Tobias_Holzmann/publication/307546712_Mathematics_Numerics_Derivations_and_OpenFOAMR/links/58acbe3992851c3cfda05ab4/Mathematics-Numerics-Derivations-and-OpenFOAMR.pdf

A huge variety of sample proofs is given in

https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-143.pdf

Demonstation: if a research problem in some form of experimental science is described mathematically, then it makes sense to require some form of demonstration of the correctness of a proposed solution. 

Examples of physics demonstrations are given in

https://www.researchgate.net/publication/256120711_Some_simple_physics_demonstration_experiments

Conference Paper Some simple physics demonstration experiments

Book Mathematics, Numerics, Derivations and OpenFOAM®

Dear Prof. James F Peters

Thank you very much for your answer and the .reading materials,

Regards,

Anup

Dear Prof. James F Peters,

Your first suggestion needs no explanation. In the second part, if I understood correctly, you are suggesting to make experimental validation of the mathematical model. .To do so we would require numerical values of the relevant physical variables. To get that, two different approaches are possible. We can simulate the physical system by using the mathematical model with  available numerical technique or we can find an analytical solution that can then be used to get the required numerical values.

For example, should we at the first instance use the computational electromagnetic tools or try to use  standard analytical methods to solve a  problem related to a well shaped microwave radiator?. .   .

@Anup Kumar Bandyopadhyay: [For an experimental science], we can simulate the physical system by using the mathematical model with  available numerical technique or we can find an analytical solution that can then be used to get the required numerical values.

Yes, I agree.   Simulation of a physical system works well, provided the particular simulation accurately reflects a physical system.   That is. with a proposed simulation, we need to be be assured that the principal features of a physical system are adequately represented in the engine that drives a simulation.

Excellent post Dr. James F Peters , recommended.

Emad

Dear Prof. Peter,

"That is, with a proposed simulation, we need to be  assured that the principal features of a physical system are adequately represented in the engine that drives a simulation. "

This calls for the validation of the simulator itself. In case of any discrepancy in the result we may not be sure whether it is due to wrong mathematical modeling or error in the simulator.

From a logical perspective, if one is building a model of a problem as a tool which should provide an answer to that problem, or to describe the nature of that problem...one needs to be certain that the model is an accurate approximation of the problem itself.  Otherwise the 'answer' will be wrong in the larger context of the inital question.