FUNDAMENTAL SOLUTIONS TO FRACTIONAL KDV EQUATION WITH CAPUTO-FABRIZIO DERIVATIVE

The concept of derivative is used to describe the rate of change of a given functionand so we can use it to model the systems occurring in real world phenomena. Due tothe complexity of problems in engineering and scienti c disciplines, the de nition ofinteger order derivative extended to the concept of fractional derivative. Comparedwith integer order, a signi cant feature of a fractional order di erential operator is itsnonlocal behavior. In other words, the future state of a process described by fractionalderivative depends on its current state as well as its past states. Therefore, by makinguse of di erential equations of arbitrary order, one can describe the memory andhereditary properties of various important materials and systems. So, in recent yearsfractional di erential equations have been paid of great interest and there appearednew areas for applications of initial and boundary value problems of such equations.Started by Riemann-Liouville fractional derivative, some new de nitions for derivativeof arbitrary order have been updated. Recently, the new Caputo-Fabrizio fractionalderivative, with no singular kernel, was proposed in [1], and further studied in [2].Caputo-Fabrizio fractional derivative has many signi cant properties such as its abilityin describing matter heterogeneities and con gurations with different scales. It isknown that, the representation of solutions for initial-boundary value problems ofdi erential equations in an explicit form is important but, not always easily obtained.In this regard, the fundamental solutions (or Green functions) represent an importanttool, since knowledge of these functions allows us to obtain analytical representationsof solutions of initial-boundary value problems. In the present paper we consider thenonlinear KdV equation with time fractional Caputo-Fabrizio derivative of the form(0.1) cD t (u(x; t)) = 6uux - u(xxx); 0 < α < 1;under the initial condition u(x; 0) = g(x). Using the Laplace transform with respect to the temporal variable t and the exponential-Fourier transform with respect to x-variable, we obtained the fundamental solu-tions of the proposed problem. It is important to note that, by using the newtime-fractional derivatives, the fundamental solutions are expressed by elementaryfunctions and Bessel functions. This is an important advantage over the using Ca-puto time-fractional derivatives, because, the fractional derivative with singular kernelleads to fundamental solutions expressed with Mittag-Le er functions.

Keywords:

Fundamental solution , fractional KdV equation , Caputo-Fabrizio derivative