The benefit of fractional calculus lies in finding new additional solutions to differential equations in addition to the solutions that result from ordinary calculus. Also, there are some mathematical problems that are formulated in the form of fractional differential equations, which incrementally need to be separated and fractional integration to be solved. It is also natural to obtain by using the two methods the same results to solve a particular problem.

Moreover, due to the effective memory function of fractional derivative, fractional differential equations have been widely used to describe many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic mode.

Dear Maged G. Bin-Saad

Thank you for your reply. But It is not because fractionnal partial differential equations have been widely used to describe many physical phenomena that it was a good idea and that we cannot do otherwise, with a greater physical sense and without the drawbacks associated to fractional operators (physical inconsistence among others).

Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.

http://www.scholarpedia.org/article/Partial_differential_equation

From the mathematical standpoint it is surely useful to introduce the mentioned equations. When we have a class of processes with a formalized peculiarity isn't it good to have more classes of models, which structure can capture such peculiarity? Perhaps, I'm rather a computational scientist, more attention should be paid on problems whether some classes of models are identical (within some error level, maybe) or what is the best class of models in the specific case?

Dear Aparna Sathya Murthy

Thank you for your reply.

But the paper

Article Some Alternative Solutions to Fractional Models for Modellin...

shows that some classes of classical partial differential equation with spatially variable coefficients have the same behaviour as fractional partial differential equations, but without drawbacks associated to fractional opertors.

Thus, what allow fractional PDE, that do not allow the classical partial differential equation with spatially variable coefficients ?

As alluded to earlier, fractional partial differential equations (fPDEs) provide a useful tool to describe anomalous diffusion (super diffusion and subdiffusion). However, the main advantage of fPDEs is that the spatial variation of the coefficients need not be known apriori to utilize the classic PDEs. This is important in subsurface flow problems where the reservoir width spans a considerable range. How do you effectively characterize this variation effectively? Existing methods introduce significant uncertainty and contribute to the disparity between observed measurements and predicted measurements. As such fPDE allow us to circumvent some of these challenges.

Therefore accurate parameter estimation of the fractional exponent(s) and coefficient(s) is essential to the widespread adoption of fPDEs in the real world.

Dear Abiola D. Obembe

Thank you for your interesting reply.

As shown in

Article Beyond the particular case of circuits with geometrically di...

with the PDEs proposed , you have to find the same number of parameters as for a fPDE. So the apriori knownledge of the spatial variations is not useful. On the other hand, in a modelling approach, it is often more interesting to get a model with a physical meaning (thus with consistent spatial coeficients) than a black box such as a fPDE.

Если расмотрим тел Кельвина-Фойхта. Тогда связь между скоростью деформации и напряжением определяется соотношением, к примеру, предположим наличие масштабной инвариантности распределения релаксационных параметров. Таким образом, наличие временной масштабной инвариантности приводит к необходимости использования реологических моделей в дробных производных. Реологический закон с дробными производными для модели, включающей всего лишь различные пружины и вязкие элементы, например постулируется существование самостоятельного типа деформации - высокоэластичной деформации, которая не может быть сведена к сумме упругости и вязкого трения.

(If we consider the bodies of Kelvin-Voigt. Then the relationship between the strain rate and stress is determined by the relation, for example, assume the presence of scale invariance in the distribution of relaxation parameters. Thus, the presence of time scale invariance leads to the need to use rheological models in fractional derivatives. A rheological law with fractional derivatives for a model that includes only various springs and viscous elements, for example, the existence of an independent type of deformation is postulated - highly elastic deformation, which cannot be reduced to the sum of elasticity and viscous friction.)

If we don't have geometrical meaning, non-locality in fractional derivatives

and independency of function then fractional partial differential equations

means what?

Dear Usmonali Saydullayev

In the example you propose, you have a fractional behaviour. But a fractional behaviour can be modelled by many other tools then fractional models. See

Article Fractional Behaviours Modelling with Volterra Equations: App...

as a recent example of what I want to say. Fractional behaviours are also permitted by partial differential equations (integer ones) with spatially variable coefficients:

Preprint Beyond the particular case of circuits with geometrically di...

It is not useful to introduce fractional partial differential equations that suffer from the same drawbacks as other fractional models. Generally, fractional models are not essential.

Article Modelling Fractional Behaviours Without Fractional Models

I believe that such equations does not have at this point a real application to model any phisical system more than classical partial differential equations do. But there are a lot of room in the future to find out that they will become useful.

Sorry for my poor english.