In so many problems our analytical methods seems to failed to find the solution.

Aanlaytical method have limitations in case of nonlinear problem in such cases numerical methods works very well.

yes and numerical method gives us approximate solution not exact solution

Highly non linear equation are not possible to solve with anylytical techniques

Analytical method often threaths the problem by simplifications of the reality. for instance: when you have a very complex geometry and/or loading conditions, it can be easily solve by finite element method rather than an analytical approach.

Numerical method always works with iteration. When we determine the final answer for each question must together with some errors. While analytical methods the final answer is straight forward. For an example when we solve the integration using numerical methods plays with simpson's rule, trapezoidal rule etc but then analytical is integration method.

As the others indicated, many models simply have not been solved analytically, and experts believe this is unlikely to happen in the future. Numerical methods provide an alternative. True, one sacrifices some accuracy on the computation, but, on the other hand, retains the accuracy (which comes at the cost of complexity) of the model.

In fact, the absence of analytical solutions is sometimes *proved* as a theorem. See, for example, the introduction to Alekseev's book "Abel's Theorem in Problems and Solutions." http://www.nairanalytics.com/abel.pdf

First of all, it should be emphasised that the "numerical approach" is not automatically equivalent to the "approach with use of computer", although we usually use numerical approach to find the solution with use of computers. That is because of the high computer performance incomparable to abilities of human brain. Numerical approach enables solution of a complex problem with a great number (but) of very simple operations. It is perfect for the computer which is basically a very fast moron :-)

I think that we can distinguish two main situations when numerical methods are used instead of analytical methods:

1. When analytical solution of the mathematically defined problem is possible but it is time-consuming and the error of approximation we obtain with numerical solution is acceptable. In this case the calculations are mostly made with use of computer because otherwise its highly doubtful if any time is saved. It is also indivually to decide what do we mean by "time-consuming analytical solution". In my discipline even very simple mechanical problems are solved numerically simply because of laziness...

2. When analytical solution is impossible, which was discussed by eg. Alexander Sadovsky. This means that we have to apply numerical methods in order to find the solution. This does not define that we must do calculations with computer although it usually happens so because of the number of required operations.

Agniezska, I agree and thank you for adding to and modifying what I wrote. (I am sorry to hear that your field is so affected by laziness.:) I would only add that, besides the large required number of operations, I would also identify another, more qualitative, obstacle: lack of insight into the object we are trying to study. Not sure if such insight can always be obtained by doing sufficient operations; I'd think, sometimes, it is the physics behind the phenomenon that eludes the researcher.

Here come to the philosophical question: The world is so complex, then why do we "need" the model problem?

We realize why then we can appreciate the beauty of analytical approach.

Comparing analytical method with numerical method is like comparing orange and apple. They serve for different purposes. Analytical method is to understand the mechanism and physical effects through the model problem. It is also useful to validate the numerical method. The numerical method is mainly to solve complex problem, physically or geometrically.

I understand nowadays ppl use numerical software extensively to many complex problem. In university, probably most students don't write numerical code to solve problem except for control engineering. But, we should bear in mind that all the software we currently use have been validate using the analytical solution already.

A closed form solutions can be existed for the problems with more assumptions solved by analytic method (calculus) whereas an approximate solutions can be obtained for the complex problems (i.e) stress analysis for aircraft wing solved by numerical method with negligible error.

Deivanathan, I wouldn't make the generalization that numerical methods are simple. Simplicity is, of course, subjective, but compare the method of lines to Finite Elements. The latter requires advanced functional analysis, while the former can be easily implemented with an elementary knowledge of calculus alone.

I agree with Dr. Shiun-Hwa’s opinion. An analytical or closed-form solution provides a good insight in phenomena under the question. Numerical methods often give a clue what kind of closed-form solution could be achieved. In my way I always look for understanding of a problem, so I prefer, whenever possible, the quest for a formula.

Numerical methods can solve real world problems, however, analytical solutions solve ideal problems which in many cases do not exist in reality.

See below is a link for simple problem solved analytically and numerically:

The link below shows the theory:

https://www.researchgate.net/publication/237050780_Solving_Ordinary_Differential_Equation_Numerically_(Unsteady_Flow_from_A_Tank_Orifice)?ev=prf_pub

The link below shows the Excel sheet model for both analytical and numericall solutions.

https://www.researchgate.net/publication/237050796_Solving_Tank_Problem

Data Solving Ordinary Differential Equation Numerically and Analy...

Data Solving Water Flow Through an Orifice in the Bottom of a Tank

If there is a possibility to get the solution analytically and numerically then prefer the analytical solution. First, the analytical solution is THE solution for multitude of particular cases, while the numerical solution has to be obtained anew for each such case separately. Another thing is tthe undestanding of inner work of any given numerical algorithm, its accuracy and applicability. Don't trust the computer too much, see the example (Siegfried M. Rump, 1988):

Given a pair of numbers (a,b) = (77617, 33096) compute

f = 333.75b^6 + a^2*(11a^2b^2 - b^6 -121b^4 -2) + 5.5b^8 + a/(2b)

Most likely you will obtain f=1.172603 (in single precision) and similar result in double and quadruple precision. Yet the true value is f = -54767/66192, i.e.

approximately f = -0.82739605994682135, where the last digit is uncertain within 5 units.

Are you sure you can trust your numerical solution (specifically in the case of differential equations) when rounding errors are out of control? With millions of intermediate results, like in finite element methods?

Suppose you have a mathematical model and you want to find a solution to the set of equations in order to understand its behavior. There are three situations to approach the solution depending on your set of equations:

1-The best case is when you can use simple math techniques such as trigonometry or calculus to write down the solution. This gives you an exact solution of how the model will behave under any circumstances. This is called the analytic solution, because you used analysis to figure it out. It is also referred to as a closed form solution.

2-For more complex models, the math becomes much too complicated. Then you turn to numerical methods of solving the equations. For a differential equation that describes behavior over time, the numerical method starts with the initial values of the variables, and then uses the equations to figure out the changes in these variables over a very brief time period. Its only an approximation, but it can be a very good approximation under certain circumstances.

3-There are also models for which it is not possible to find an analytical solution.These are models that have non-linear equations. For these models there are methods such as the perturbation method which can be used to find an approximate analytical solution within a certain range. The advantage here over a numerical solution is that you end up with an equation (instead of just a long list of numbers) which you can gain some insight from.

It is always a good thing to at least try to find an analytical solution. Surely, non-linear equations may be tricky, but you are sure that x^2+1=0 has no real solutions while many numerical methods will give you the approximate solution, namely x=0. In case when your complicated equation has more than just one solution, the numerical solver will usually produce only one answer for you. Not necessarily the most appropriate/interesting one.

There is a special case, called 'data fitting' (="solving the equation system with more equations than there are unknowns", and when additionally the fitted data are uncertain). Here, in classical sense, the solution simply doesn't exist. To get valuable results anyway, we switch to solve a different problem, closely realted to our original system of equations. Linear, unconstrained problems aside, the numerical solver is the only choice.

I think both methods are relevant and are great to use. Especially the numerical method FEM is a excellent tool to solve complicated geoemtrical shapes with a boundary and load condition that is diffulcult to describe with analytical experissons available in the industry!

I thin kthe best thing is to combine accurate and reliable experimental testing with a simple to use anaytical expression of the involved physics and mechanisms and complement with a numerical FEM-model where a set of parameters can be adjusted and changed with the aid of Design of Experiments.

There are generally three aproches by which scientific problems/equations are solved :

Analytical,Numerical and Experimental.However,we cannnot perform experimental method every time because of cost and time constraints.Analytical methods are the conventional methods to solve problems.But due to restraints caused due to complex Geometry,Boundary conditions,etc we are not able to solve equation analytically.So,from past many years we are moving towards Numerical methods as they can provide almost accurate result in comparision with Analytical method and that too in less time and easy way.

Famous Navier-stoke equation has not been solved till now analytically but can be easily solved by Numerical Schemes.

Analytical solutions are exact solutions while numerical ones are approximatives. In many cases, we cannot find analytical solutions for solving problems encountred in pratice and then the governing equations must be solved numericaly in spite of the approximative approach. In addition, in numerical methods the solution of problem must be validated experimetally or by others works from the literature.

In 1970's computers and numerical methods changed everything in research.

The best thing that numerical methods did is to solve nonlinear systems of equations.

Many problems exist that have no analytical solution. And even problems with analytical solutions do have them because lots of constants are assumed to be constant. However the analytical solution to a simplified problem learns us a lot about the behavior of the system. On the other side if no analytical solution method is available then we can investigate problems quite easily with numerical methods. However care has to be taken that a converged solution is obtained. This means that you have to research wether your step sizes are small enough to find the solutions of the equations you try to solve.

However this is not necessarily always true. Suppose if a company wants to know the trend of the results if they change a certain parameter and computational power is limited. Then you might not require full convergence. This is often the case in fluid dynamic problems in which the equations are not exact and models play a role.

Hope this helps

Actually both solutions are needed. However, there are many problems do not have analytical solutions. In this case you are obliged to find the solution numerically. But you should be careful about stability conditions and accuracy.

Note also that if analytic solutions are available you can use them as benchmarks for the numerical methods.

Most of the points are already stated above. Let me summarized them here

(i) There are many problems where it is known that there is an analytic solution(existence). But we do not know or can not find it in the closed form. For example normal distribution integral.

(ii) There are many problems where solutions are known in closed form which is not simple or it is in the form of an infinite series where coefficients of the series are in the form of integrals which are to be evaluated. Examples are Fourier series, Laplace transform or Fourier transform based methods. These solutions do not give any insight of the problems. Most of the non-linear problems exhibit this nature.

(iii) Numerical methods became popular because of computers but they existed before computers came into being such as Newton-Raphson method, Newton- Cotes formulae, Gaussian Quadrature etc.

(iv) There are application where you want to have real-time solution, that is , you not find solution as quickly as possible so that further decision can be taken. In such cases efficient Numerical Methods are applicable. Examples are in Space Science and Bio Science.

See, for example, this paper

http://arxiv.org/pdf/cs/0701192v5.pdf

Here is what Lewis Fry Richardson said in 1908 about the relative merits of analytical and numerical solutions:

Further than this, the method of solution must be easier to become skilled in than the usual methods (i.e. analytical solutions). Few have time to spend in learning their mysteries. And the results must be easy to verify—much easier than is the case with a complicated piece of algebra. Moreover, the time required to arrive at the desired result by analytical methods cannot be foreseen with any certainty. It may come out in a morning, it may be unfinished at the end of a month. It is no wonder that the practical engineer is shy of anything so risky (Richardson 1908).

hello,

Finding an analytical solution for the highly nonlinear problem is not possible one. At that time you should go for numerical method to solve that equations.

Rough summary from Partial Differential Equations:

analytical solution for boundary value problem is possible

1. Linear PDE

2. Simple geometry of the domain: Rectangle, Cube in Cartesian, Cylindrical or Spherical coordinate system and a few other geometry

3. Homogeneous boundary conditions (same along coordinate line)

If in the case of Cartesian coordinate - basis (taken in Hilbert space) consists of sin cos sinh cosh and their combinations, then in Cylindrical cs one needs already all types of Bessel functions. This is highly sophisticated task. It may happen that Fourie series solution is though analytically correct but will require very lengthy computation due to embedded Eigen value problem with Bessel function etc etc

All this only for simple geometry!!!

IF SOMETHING 1, 2, 3 is not fulfilled then the solution is in general not possible with some exeptions.

That is why NUMERICAL METHODS ARE EXISTING!

E.g. Problem - deformation of a body of arbitrary geometry - only numerical solution (eg FEM) is possible even for the linear problem.

Of course, as mentioned already, all set of analytical solutions are perfect basis for the verification of the numerical method

A major advantage of numerical method is that a numerical solution can be obtained for problems, where an analytical solution does not exist. An additional advantage is, that a numerical method only uses evaluation of standard functions and the operations: addition, subtraction, multiplication and division. Because these are just the operations a computer can perform, numerical mathematics and computers form a perfect combination.

MOST OF THE PROBLEMS WE ENCOUNTER DO NOT HAVE ANALYTIC SOLUTION AND WHERE THEY EXIST, IT INVOLVES MUCH COMPUTATIONS. THAT HAS LED TO THE EMERGENCE OF MANY NUMERICAL METHODS

Analytical methods are limited to simplified problem. However numerical methods are used for practical problems. See these links that may help. It shows analytical and numerical solutions to several problems:

https://www.researchgate.net/publication/266266232_A_fully_distributed_spreadsheet_modeling_as_a_tool_for_analyzing_groundwater_level_rise_problem_in_Jeddah_city

https://www.researchgate.net/publication/230726475_Influence_of_temporal_fluctuations_and_spatial_heterogeneity_on_pollution_transport_in_porous_media

https://www.researchgate.net/publication/232946064_Analysis_and_numerical_simulation_of_a_singlewell_tracer_test_in_homogeneous_layered_and_slightly_tilted_formations

https://www.researchgate.net/publication/230720658_Simulation_of_solute_transport_under_oscillating_groundwater_ow_in_homogeneous_aquifers

Article A fully distributed spreadsheet modeling as a tool for analy...

Article Influence of temporal fluctuations and spatial heterogeneity...

Article Analysis and numerical simulation of a single-well tracer te...

Article Simulation of solute transport under oscillating groundwater...

For every ordinary differential equations can not have exact solution. In this cases numerical methods play crucial role. 

Analytical methods are more effective when dealing with linear differential equations, however most non-linear are too complex and can only be solved using these numerical methods

Numerical answers are easier to find! If you can find an analytical answer it is always preferable!

These methods can help each other.

It will be a difficult task to find the analytical solution for complex problems. you have to deal with higher order PDEs. Numerical methods give approximate solutions and they are much easier when compared to Analytical methods.

This may help you:,

Conference Paper Comparison between an analytical method and two numerical me...

When analytical solution is impossible, this means that we have to apply numerical methods in order to find the solution.

Generally, analytical solutions are possible using simplifying assumptions that may not realistically reflect reality. In many applications, analytical solutions are impossible to achieve. Numerical methods makes it possible to obtain realistic solutions without the need for simplifying assumptions. Additionally, analytical solutions can not deal with discrete data such as the dynamic response of structures due to Earthquakes.

When analytical approaches do not lead to a solution or are too time-consuming numerical methods are far more efficient. While there is always criticism on the approximation that results from numerical methods, for most practical applications answers obtained from numerical methods are good enough. After all didn't most of us use 22/7 to approximate pi while doing problems in our middle schools?

Good question, really useful answers, I agree with Dr.Anastas Ivanov Ivanov

Analytical methods, if available, are always the best. However, these are impossible to achieve in some cases. Hence, numerical methods gives as the possibility to obtain an approximate numerical solution.

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.

The limitations of analytic methods in practical applications have led scientists and engineers to evolve numerical methods.There are situations where analytical methods are unable to produce desirable results. Even if analytical solutions are available, these are not amenable to direct numerical interpretation.

Numerical methods just evolved from analytical methods... Just remove manual intervention of human by using computers.

For example, to find integral of function 'f(x)' containing trigonometric, exponential, power terms, etc. requires lots of understanding of maths...

In numerical methods, it's just Area under the curve 'f(x)' which can be calculating by adding number of rectangular strips...

Also, it's helps in solving more complex problems using higher memories and speed of computers...

The numerical methods are used for deeper understanding to predict the anomalies which are not possible in the analytical methods because the analytical method can solve only two or three unknown variables but numerical methods can do much more than it very accurately.

Numerical method of solution to Mathematical problems will be preferable over the analytic counterpart if;

1) the problem fails to have a closed-form solution,

2) the problem become well-posed in the limiting sense,

3) even when closed-form solution exists but lack computational merit,

4) the solution method is unnecessary lengthy.

1. Numerical methods offer approximation solutions to Mathematical problems where

i) analytical methods of solutions may not exist,

ii) data available does not admit the applicability of the direct use of the existing analytical methods.

2. Where existing analytical methods turn out to be time-consuming due to large data size or complex functions involved, Numerical methods are used since they are generally iterative techniques that use simple arithmetic operations to generate numerical solutions.

. Numerical methods offer an approximation of solutions to Mathematical problems where: i) analytical methods of solutions may not exist, and ii) data available does not admit the applicability of the direct use of the existing analytical methods. 2. Where existing analytical methods turn out to be time-consuming due to large data size or complex functions involved, Numerical methods are used since they are generally iterative techniques that use simple arithmetic operations to generate numerical solutions.

• Numerical Methods are mathematical way to solve certain problems.Whether the equations are linear or nonlinear, efficient and robust numerical methods are required to solve the system of algebraic equations.
• Analytical solutions are exact solutions based on mathematical principles.
• However, the governing partial differential equations of fluid flow are complex and cannot be solved by analytical means. The partial differential equations are therefore converted into a system of algebraic equations that are subsequently solved through numerical methods to provide approximate solutions to the governing equations.
• Although we rarely reach on exact answer , we can get really close to the exact answer much quicker than solve analytically.

https://youtu.be/F2NWYEOHIkc

- Analytical methods offer the 𝐞𝐱𝐚𝐜𝐭 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧 (which is not always possible to obtain) at the expense of the long time spent in computing.

- Numerical methods solve the problem of 𝐭𝐡𝐞 𝐥𝐨𝐧𝐠 𝐭𝐢𝐦𝐞 𝐬𝐩𝐞𝐧𝐭 𝐢𝐧 𝐜𝐨𝐦𝐩𝐮𝐭𝐢𝐧𝐠, but offer only an 𝐚𝐩𝐩𝐫𝐨𝐱𝐢𝐦𝐚𝐭𝐢𝐨𝐧 of the 𝐞𝐱𝐚𝐜𝐭 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧.

- Numerical methods prove to be useful when one is able to prove the existence of the solution theoretically without being able to obtain its analytical form, at this moment, one usually turns to the numerical methods to get an approximation of the solution which robustness depends on the numerical method used.

For example, solving the equation f(x)=0, where f is a polynomial of degree 5 or up. Here, the existence of the exact solution is due to the fundamental theorem of algebra and we can obtain the approximation of the exact solution using numerical methods such as the bissection method, Newton method ...

The best way to solve a model that represents a problem in real life is to use the numerical approach. Because reality is complex and relative to us. I consider it scientifically flawed to compare the results of the numerical approach with the results of the analytical methods. For comparison the best way to solve is to use a calibrator eg RMSE etc.

Numerical approach gets the advantage of the repetitive nature of computers to solve complex problems that require lots of iterations. Numerical analysis applications are distributed among different areas, such as computational fluid dynamic analysis, Structural analysis, Kinematic Analysis, and even economic analysis. Numerical methods is the working principle behind Finite-element-analysis, Simulations, and many more.