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On the use of Mohand integral transform for solving fractional-order classical Caputo differential equations

Qureshi Sania, Yusuf Abdullahi, Aziz Shaheen

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

99--109

EN

In this research study, a newly devised integral transform called the Mohand transform has been used to find the exact solutions of fractional-order ordinary differential equations under the Caputo type operator. This transform technique has successfully been employed in existing literature to solve classical ordinary differential equations. Here, a few significant and practically-used differential equations of the fractional type, particularly related with kinetic reactions from chemical engineering, are under consideration for the possible outcomes via the Mohand integral transform. A new theorem has been proposed whose proof, provided in the present study, helped to get the exact solutions of the models under investigation. Upon comparison, the obtained results would agree with results produced by other existing well-known integral transforms including Laplace, Fourier, Mellin, Natural, Sumudu, Elzaki, Shehu and Aboodh.

2

Laplace-Carson integral transform for exact solutions of non-integer order initial value problems with Caputo operator

Kumar Prem, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 1

57--66

EN

Finding the exact solution to dynamical systems in the field of mathematical modeling is extremely important and to achieve this goal, various integral transforms have been developed. In this research analysis, non-integer order ordinary differential equations are analytically solved via the Laplace-Carson integral transform technique, which is a technique that has not been previously employed to test the non-integer order differential systems. Firstly, it has proved that the Laplace-Carson transform for n-times repeated classical integrals can be computed by dividing the Laplace-Carson transform of the underlying function by n-th power of a real number p which later helped us to present a new result for getting the Laplace-Carson transform for d-derivative of a function under the Caputo operator. Some initial value problems based upon Caputo type fractional operator have been precisely solved using the results obtained thereof.

3

Using Shehu integral transform to solve fractional order Caputo type initial value problems

Qureshi Sania, Kumar Prem

Journal of Applied Mathematics and Computational Mechanics

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2019

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Vol. 18, nr 2

75--83

EN

In the present research analysis, linear fractional order ordinary differential equations with some defined condition (s) have been solved under the Caputo differential operator having order α > 0 via the Shehu integral transform technique. In this regard, we have presented the proof of finding the Shehu transform for a classical nth order integral of a piecewise continuous with an exponential nt h order function which leads towards devising a theorem to yield exact analytical solutions of the problems under investigation. Varying fractional types of problems are solved whose exact solutions can be compared with solutions obtained through existing transformation techniques including Laplace and Natural transforms.

4

On nonlinear neutral Liouville-Caputo-type fractional differential equations with Riemann-Liouville integral boundary conditions

Ahmad Bashir, Ntouyas Sotiris K., Alsaedi Ahmed

Journal of Applied Analysis

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2019

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Vol. 25, nr 2

119--130

EN

This paper studies neutral Liouville-Caputo-type fractional differential equations and inclusions supplemented with nonlocal Riemann-Liouville-type integral boundary conditions. Sadovskii’s fixed point theorem is applied to establish the existence result for the single-valued case, while the multivalued case is investigated by using nonlinear alternative for contractive maps. Examples are constructed to illustrate the main results. The case of nonlinear nonlocal boundary conditions is also discussed.

5

Fractional integral inequalities for composite and k-composite preinvex functions

Kashuri A., Liko R., Iqbal S.

Fasciculi Mathematici

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2019

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nr 62

57--79

EN

In this article we found some Ostrowski inequalities for composite and k-composite preinvex functions via fractional integrals. Also some special cases will be given.

6

Some new fractional Fejér type inequalities for convex functions

Set E., Mumcu Î.

Fasciculi Mathematici

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2017

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Nr 59

145--157

EN

In this paper, firstly, a new identity for conformable fractional integrals is established. Then by making use of the established identity, some new fractional Fejér type inequalities are established. The results presented here have some relationships with the results of Set et al. (2015), proved in [6].

7

Hermite-Hadamard type fractional integral inequalities for generalized (r; g,s,m,φ)-preinvex functions

Kashuri A., Liko R.

Fasciculi Mathematici

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2017

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Nr 59

43--55

EN

In the present paper, a new class of generalized (r; g, s, m, ϕ)-preinvex functions is introduced and some new integral inequalities for the left hand side of Gauss-Jacobi type quadrature formula involving generalized (r; g, s, m, ϕ)-preinvex functions are given. Moreover, some generalizations of Hermite-Hadamard type inequalities for generalized (r; g, s, m, ϕ)-preinvex functions via Riemann-Liouville fractional integrals are established. These results not only extend the results appeared in the literature (see [1],[2]), but also provide new estimates on these types.

8

Hermite-Hadamard type inequalities for h-preinvex mappings via fractional integrals

Matloka M.

Control and Cybernetics

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2015

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Vol. 44, no. 2

275--285

EN

In this paper, we ﬁrst establish the Hermite-Hadamard inequalities involving Riemann-Liouville fractional integrals for the h-preinvex function. Then, some Hermite-Hadamard type integraf inequalities for the fractional integrals are obtained.

9

On the set of solutions of fractional order Riemann-Liouville integral inclusions

Abbas S., Benchohra M.

Demonstratio Mathematica

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2013

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Vol. 46, nr 2

271--281

EN

In this paper, we prove the arcwise connectedness of the solution set of a nonclosed, nonconvex Fredholm type, Riemann–Liouville integral inclusion of fractional order.