A Review of Generalized Fractional Integral Operator in Fractional Differential Equations

A new contemplate fractional integral operator employing K4 mapping is the focus of this chapter. For the suggested generalized operator, the Mellin and Laplace transforms as well as the research of boundedness and the development of novel composite qualities are also addressed. K4 mapping and Hilfer differentiations are utilized to decipher the fractional derivative equation based on the results reached. Using the K4 mapping as a generalization of the M-series, we may conclude that various outcomes described before are gladly followed as specific findings in our inquiry. Recent research in applied sciences, engineering, and technology has discovered new uses for generalized special functions. These conclusions must be "validated" by the establishment of several corollaries and lemmas. Several fractional integral and differential operators have been investigated in detail due to the widespread use of fractional calculus by researchers including Kilbas, Kalla, Khan, McBride, Kumar and Saxena, and Kiryakova, amongst others. It was Sharma who first created the K4 function.: where are the Pochhammer notation. The above equation (1) is only valid when none one of the variables j b is zero or integer (negative) and if any num. variable i a is zero or negative integer, the series changes to a polynomial in variable x. The series is convergent if q +1< p . If we replace , in (1), we obtain the subsequent result: where is the well known generalized M-series which was explored by Sharma and Jain, is denoted as power series: Pochhammer symbols. The series given in (3) is described as only when none of the variable , is zero or a negative integer; if any numerator variable i a is zero or a negative integer, then the series changes to a polynomial in z. If , The equation (3) is confluent for all variable z, if p = q +1, it is concurrent for

= q + 1 and , equation will converge on conditions based on varibles values. Prabhakar introduced the generalized Mittag-Leffler mapping which may be found from (3) for ; as The induced M-series represent by (3) can be revealed as a particular case of Wright generalized hypergeometric function and Fox H-function as together with the convergence conditions as known by Mathai, Braaksma, Kilbas and Saigo, Ram J. and D. Kumar. Wright given generalized hypergeometric function by resources of the sequence representated in the type given as Hilfer gave fractional differential operator with two variables of order in the form as mantion below:

If and are the Pochhamer notations, the functions represented by (1) can easily be expressed by the Mellin-Barnes types integral as given below: The goal of this section is to find different qualities like Mellin and Laplace transform of the suggested extended integral operator with function written as follows: where .The operator was studied by Shukhla and Prajapati. It includes special cases.

On setting , equation (12) it capitulates the following operator linked with Mittag Leffler function studied by Kiryakova.

Proof: The Mellin transform as defined follows: and using eq.(11), we obtain left hand side of (14) By replacing the order of integral, as admissible under the same conditions stated in theorem If we replace x = t + u on the R.H.S. of the equation (4.16), we obtain At the present on indicating K4 function in reference of its Mellin-Bernes contour integration of equation (10), we see While using following formula to solve u‘s integral at right hand side in equation (17) Now by goodness of definition of H function in terms of Mellin-Bernes contour integration(6, 7) resist the desired results (14).

On settingequation (14) yields the following result

is Wright function as given by eq.( 8). Proof: By applying the result (11), we obtain by replacing the order of integration, admissible under the same circumstances declared in theorem 19 If we replace x =t + u on the R.H.S. of the equation (20), we obtain while using series definition of K4 function (1), we see which completes the proof of theorem where is represent the Laplace transform of and k has same meaning as defined in (19).

If the conditions mentioned in the equation (4.11) is satisfied and then Proof: On making use of equation (1) and (11), we see L.H.S. of equation (22) While taking use of series definition of Wright function (8) yields the result (22) after simple modification.

With all constraints as stated in equation (11), and the mapping g be the nth space L(a,b) of the Lebesgue measurable mappings on limited interval [a,b] (a < b) of the real line is defined by The integral defined by (11) is given as where B is given by Proof: By using of (24) in L.H.S. of (25), we obtain

On using series expressed as K4 function (1), we have This completes the proof of result shown by equation (26).

With all limitations on variables as given in equation (19) the solution of the Fractional derivative equation Proof: Put in application the Laplace transform to both side of (27) and applying Convolution theorem of the Laplace transform On imitating inverse Laplace transform and definition of Wright function Which yields proof of theorem as required.

In this chapter, we created a generalized integral operator that incorporates functions as well. In many physical science equations, such as the diffusion equation, fractional kinetic equation, vibrations, etc., numerous abstract findings based on differential operator theory may be deduced from these results. The Mellin and Laplace transforms have been examined using the proposed fractional integral operator. Eventually, these assumptions are utilized to solve fractional derivative equations, including mapping and associated with Hilfer's differentiations. Assumptions are typically made in many ways in the fractional integral operator of particular mappings.

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Research Scholar, Sunrise University, Alwar, Rajasthan