It is useful to transform a differential equation into a difference equation when an approximate solution is needed. If you want analytic solution then difference equation can give you idea as to what type of solutions are there.

Dear Dr. T. F. Ibrahim,

Yes, since we face simultaneously both kinds of systems discrete and analog.

Often differential equations are difficult to solve and so various numerical schemes are adopted to obtain an approximate solution. Difference equation is one the simplest approach to break down the analytical problem. Differential equations will always give a complete and exact solution while difference eqns will tend to approach the exact solution as close as possible! 

Dear Prof  R. C. Mittal

Dear Prof Afaq Ahmad

Dear Prof Vikash Pandey

 A lot of thanks for your answers .

 Best Regards ,

Dr/Tarek Ibrahim ( T. F. Ibrahim )

Associate professor

¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia

²Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

E-mails: [email protected]

[email protected]

[email protected]

Personal Websites :

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=A/

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=E

https://www.researchgate.net/profile/Tarek_Ibrahim2/?ev=hdr_xprf

http://scholar.google.com/citations?hl=en&user=K0uEihsAAAAJ&sortby=pubdate&view_op=list_works&pagesize=100

When we do basic modelling steps on an infinitisimal volume ( or interval ) for a physical problem say involving rate of growth,  invariably we come across  difference equation and using some short of continuum hypothesis, we get differential equation. Because, seemingly  different problems can be put under one umbrella ( grand unification: what mathematics helps to do). Even where analytical solution is available for the resulting differential equation, it may contain some terms which can not be evaluated exactly. Therefore, even in this case one resorts to numerical approximation. Then, one natural question is: why not work with difference equation  which comes from basic modelling steps ? The plausible answer to this question is : we have many more  robust and efficient difference schemes developed to solve the model problem.

One reason to prefer difference equations is that the vast majority of our computers are digital and are thus more suited to solving difference equations than differential equations that can only be solved approximately using finite difference methods such as Runge-Kutta, etc. If we all had general purpose analog computers, we would probably use them to approximate the solutions of difference equations.

  If we investigate   differential equations  on the finite interval (or  on the bounded domain) sometimes   it is useful to use numerical methods for solving differential equations. But if we investigate differential equations on the unbounded domain the numerical methods are useless. 

Dear Prof  Amiya K. Pani

Dear Prof Elman Shahverdiev

Dear Prof Gevorg Grigorian

Dear Prof Julien Clinton Sprott

 A lot of thanks for your answers .

 Best Regards ,

Dr/Tarek Ibrahim ( T. F. Ibrahim )

Associate professor

¹Current address : Department of Mathematics, Faculty of Sciences and arts (S. A.) ,King Khalid University, Abha , Saudi Arabia

²Permanent address: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.

E-mails: [email protected]

[email protected]

[email protected]

Personal Websites :

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=A/

http://eupc.mans.edu.eg/V2/get?Dr=27311031200455&T=2&L=E

https://www.researchgate.net/profile/Tarek_Ibrahim2/?ev=hdr_xprf

http://scholar.google.com/citations?hl=en&user=K0uEihsAAAAJ&sortby=pubdate&view_op=list_works&pagesize=100

Is it useful to transform problems between difference equations and differential equations ? - ResearchGate. Available from: https://www.researchgate.net/post/Is_it_useful_to_transform_problems_between_difference_equations_and_differential_equations [accessed Dec 28, 2015].