Development of new methodologies for solving nonlinear differential equations

Or

Development of new methodologies for solving strongly nonlinear oscillatory systems

I think at present the topic of the Fractional Calculus and Fractional Differential Equations is very interesting topic ,where this topic has a wide applications.

A topic of wide interest in hydrodynamic is the solution of nonlinear and dispersive wave equations, particularly strongly nonlinear waves.

Fractional calculus specially fractional Differential equation is one of important topic.

@ Abdul B.K

In my view the topic Study on

Controlled and Un -controlled vibrations through Differential Equations

is probly a topic to cover all above suggestions and many more not suggested above

B.Rath

inverse hyperbolic differential equations

Have you already chosen an advisor? If yes, it would certainly be a good idea to discuss your intended research subject with him.

For a general advice: one possibility would be to do something from the field of integrable systems.

If you are more interested in ordinary than in partial differential equations then a good place to start could be here

http://elib.mi.sanu.ac.rs/files/journals/tm/26/tm1411.pdf

For another nice and informal introduction to integrable systems see also

https://www1.maths.leeds.ac.uk/~siru/ISsurv%282015%29.pdf

There are many different approaches of integrable non linear partial differential equations to choose from: geometry, differential systems, analysis, group algebra and more. This field has past its hay days, but there are still important problems, like generalisation to more than 1+1 dimensions. Needless to say, they aren't the easiest ones.

Fractional differential equations. Katsikadelis, J. T. (2018). Numerical solution of variable order fractional differential equations. arXiv preprint arXiv:1802.00519.

Choosing a research topic is not like buying gadget from market. Everyone here will suggest you the field they are working in or field in which they are interested. You have to spend year or even more to sort out the field of research in which you can comfortably work. Otherwise you would end up hanging in between if you go with random suggestions.

And most appropriate is to ask from a professor or a supervisor who is better acknowledge with your potential and expertise.

i really agree with M. Singh. You can review the possible research lines and what is being researched in this field

A good and fast approach, it can be, simply search in Google Scholar differential equations. Browse down to the bottom of the page and you will find about 18 possible topics (Keywords) that will give you an idea of what is being published. What can interest you and above all that, which you can use for your doctoral subject is another problem. Most of the PHD programs start precisely because it is necessary to specify the research project and that can take a couple of months of review, discussion with the tutor and, if necessary, approval by the Doctoral Committee.

@Claude Pierre Massé: I'd like to complement your answer regarding integrable partial differential systems in more than two independent variables as the situation is a bit more nuanced:

1. For the case of *three* independent variables one has quite a lot of examples, both dispersive and dispersionless, using the so-called central extension technique.

2. For the case of *four* independent variables one for a long time had just a few dispersionless examples but it was recently shown that there actually are much more integrable dispersionless systems in four independent variables than it appeared, and there is a systematic construction of a new class thereof, see the paper linked below.

https://www.researchgate.net/publication/259624903_New_integrable_31-dimensional_systems_and_contact_geometry

However, the challenge of finding any examples of *dispersive* integrable systems in four independent variables is indeed still there and I would say that it is the main unsolved problem in the field of integrable systems.

It is, however, not at all clear whether nontrivial examples of dispersive integrable partial differential systems in four independent variables exist at all.

Fractional differential equations and dynamical systems.

Dear Abul Barakath K,

What type of differential equations?

Partial DE or ordinary DE, what is the approach?

Numerical or analytical?

Anyway, I suggest choosing your thesis in the domain of (Systems of linear differential equations- dynamical systems) autonomous and non-autonomous.

It includes the knowledge of several skills in mathematics:

Matrix theory, analysis, calculus of several variables, stability theory,

Lyapunov functions, modeling and more.

It is interesting and gives you the chance to be good in many domains at the same time. You need to discuss the topic with your supervisor as well.

Best regards

The best advise on possible topics for your research you'll obviously get from your supervisor(s) who will suggest you the topics, follow your progress, answer your numerous questions, guide you towards the degree and who in the first place will accept responsibility to supervise your thesis work. You may also accept advise from the members RG community who may be even more confused than you and may not be able to provide you proper guidance you need.

Rogovchenko's answer is quite clear.

@Abdul Barakath K

My suggestion is that discuss with your supervisor . If you are selecting the topic

then job will be very hard . Is it your phd or

Mphil or Msc project work ? Your background is necessary else RG suggestion is meaningless .

B.Rath

Your question is formulated too broadly and, to be quite frank, it does not mean anything. Learn well the basics of calculus, follow as many lectures as you can about ode, pde and functional analysis. Try to get yourself a vague idea about modern research in the field. Often there are introductory articles or also books dedicated to advanced topics. Take a look. You have to busted your ass.

In the end, discuss with a supervisor and slowly find your way!

@Abdul

Select the topic with your knowledge on diff

equn and application . If at present you do not have anything then work on my suggestion . However with the knowledge of supervisor else you will face problem in future.

Goodluck