Fractional Derivatives

 

Introduction:

A new definition of fractional derivatives with polished kernel which take on two different presentation for the spatial and temporal variables . The first definition show the spatial variables and second definition represented the time variable . it is helpful to apply the Laplace modified.

Fractional derivatives are the memory manager which is normally show the loss of energy,

Application and history:

In recent year the fractional derivatives investigator have expand  powerful mathematical models that use non local fractional derivatives  so these models are useful for recognize  composite system with memory effects & inborn traits. These system look in varying filed such as  chemistry , biology , signal processing, engineering, fluid mechanics and soon

We explain the main definition of fractional derivatives

Some classical rules extend  in the derivative involves chain rules, product rules ,& quotient rule . Additionally , we find results analogous  to Mean Value & roll theorem.

The first use  of fractional derivatives  was because of abel in his answer for the Tautochrone issue. The Tautochrone problem resolve with finding the shape of curve along with pendulum with same period of time.

Fractional derivatives also use in

Biophysics: which is a helpful model process with  memory effects,

Quantum Mechanics: it is helpful to understand the quantum system.

Group Application: these application use in algebraic structure

Spectroscopy: which is use to analyze spcetra.

Filed theory: These application used in engineering & physics .