Fractional Derivatives Fractional Integrals And Fractional Pdf EXCLUSIVE

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The paper is organized as follows. Section 2 presents the adopted notation. Sections 3 and 4 list the proposed definitions of fractional derivatives and integrals, respectively. Finally, Section 5 outlines some brief remarks.

The following remarks clarify the notation used in the sequel in Sections 3 and 4.(i)Let , , where denotes the real part of complex number.(ii)Let be a finite interval in , , , and .(iii)The floor function, denoted by , is defined as .(iv) is the integer part of number and the fractional part, , so that .(v).(vi) is the variable fractional order with and . is a continuous function on .(vii) is a closed contour, in the complex plane, starting at , encircling once in the positive sense, and returning to . , with and .(viii)Consider and . The so-called -gamma function, denoted by , is related to the classical gamma function by means of .(ix)The so-called -Pochhammer symbol yields .(x)The -fractional Hilfer derivative recovers, as particular cases, the fractional Riemann-Liouville derivative if and and the fractional Caputo derivative if [41].

Remark 5. In the expressions for the right and left Liouville fractional derivatives (2) and (3), respectively, some authors have a slight distinct expression, instead of just + and at the lower limit .

Remark 8. The paper does not focus on particular relations involving explicit parameters, intervals, or constants, associated with the distinct derivatives. For example, we can mention that, for , with , the Liouville fractional derivatives are of purely imaginary order. Also, for , we recover the derivative of integer order. For example, and .

In this note we present the application of fractional calculus, or the calculus of arbitrary (noninteger) differentiation, to the solution of time-dependent, viscous-diffusion fluid mechanics problems. Together with the Laplace transform method, the application of fractional calculus to the classical transient viscous-diffusion equation in a semi-infinite space is shown to yield explicit analytical (fractional) solutions for the shear-stress and fluid speed anywhere in the domain. Comparing the fractional results for boundary shear-stress and fluid speed to the existing analytical results for the first and second Stokes problems, the fractional methodology is validated and shown to be much simpler and more powerful than existing techniques. 2b1af7f3a8