# An Introduction To The Fractional Calculus And Fractional Differential Equations Pdf

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Special Functions Of Preface. Special Functions of the Fractional Calculus.

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D. More generally, one can look at the question of defining a linear operator. Fractional differential equations, also known as extraordinary differential equations, [1] are a generalization of differential equations through the application of fractional calculus. In applied mathematics and mathematical analysis , a fractional derivative is a derivative of any arbitrary order, real or complex. Therefore, it is expected that the fractional derivative operation involves some sort of boundary conditions , involving information on the function further out.

## Fractional Differential Equations, Volume 198

For this convolution, we also determine the divisors of zero. Its solution is obtained in terms of the four-parameters Wright function of the second kind. Fractional calculus of variations is one of the recent and important research areas in Fractional Calculus FC.

It is a natural extension of the calculus of variations to the case of functionals that depend not only on integer order derivatives, but also on fractional derivatives [ 3 , 23 ]. As in the case of conventional calculus of variations, a necessary condition for a function that solves an optimization problem for a functional depending on some fractional derivatives can be formulated in terms of a suitably modified Euler-Lagrange equation [ 3 ]. The Euler-Lagrange equations of the fractional calculus of variations are very special fractional differential equations that as a rule contain both the left-and right-hand sided fractional derivatives as well as their compositions.

The standard methods for analytical treatment of the fractional differential equations were worked out for equations with either only left-or only right-hand sided fractional derivatives. That is why they do not work for the fractional Euler-Lagrange equations and thus some new techniques are needed for their analytical and numerical treatment.

The case of the fractional differential equations with the left-and right-hand sided fractional derivatives defined on a finite interval has been considered in [ 14 ]. However, to the best knowledge of the authors, no methods for analytical treatment of these equations defined on an infinite interval, say, on the positive real semi-axis, are yet known the methods from [ 14 ] cannot be applied in the case of an infinite interval.

A one-parametric family of convolutions for this integral transform has been also constructed in [ 2 ]. We also refer to [ 17 ] for a general overview of the operational calculi for different fractional differential operators. Thus, the operational calculi for the fractional derivatives can be applied both for analytical treatment of the initial-value problems for the fractional differential equations and for deriving the closed form formulas for solutions of the fractional integral equations.

The rest of this paper is organized as follows. The divisors of zero of these convolutions are discussed, too. The obtained solution is expressed in terms of the four-parameters Wright function of the second kind. In this section, we remind the readers their basic properties that are used in further discussions. In the formulas 2. In what follows, we call the operators defined by 2. Because of the well-known properties:. In what follows, we shortly mention the formulas from [ 2 ] that will be used in the further discussions.

The formulas 2. Let the condition 2. The proof of the theorem is straightforward, i. Employing 2. Thus we arrived at the representation. They are given in the following theorem.

These arguments and the representation 3. Comparing the last formula with the right-hand side of the statement 3. It is worth mentioning that the initial conditions in form 4. We start with a formal derivation of a solution and then proceed with a verification that it is really the unique solution to the initial-value problem 4. Taking into account the formulas 4. Applying the property 2. Moreover, the equation 4. The representation 2.

Thus the equation 4. This equation can be easily solved with respect to the unknown function y in the frequency domain:. First we formally derive a formula for the inverse transform and then prove its validity. We start with the well-known formula. However, we formally substitute this series representation into the equation 4.

Using the formula 4. Applying this formula to the second power series at the right-hand side of 4. Later on, this function was studied in detail and employed as a kernel of an integral transform in [ 6 ]. The properties of the Wright functions of the first and the second kind are very different. For the FC applications, mainly the Wright functions of the second kind are relevant. In particular, it it the case for the problem we deal with in this paper.

Now we proceed with the first power series at the right-hand side of 4. Thus we deduced the unique formal solution to the initial-value problem 4. In the following theorem we prove that the obtained formal solution is also solution in the conventional sense. Then the initial - value problem 4.

The function y f satisfies the inhomogeneous equation 4. The proof of the theorem is a very technical one and thus we restrict ourselves to a short description of the most important steps and ideas and do not present all in part very long calculations in detail. We start with the remark that because the formal solution 4. As soon as we check that 4. Indeed, this function can be represented in form of the following Mellin-Barnes integral or the Fox H -function [ 15 ] :. Thus the function y defined by the formula 4.

To prove that the function 4. Thus we substitute 4. We start with the function F 1 defined by 4. To overcome this problem, we employ the technique of the Mellin-Barnes integrals and use the representation 4.

Finally, we use the Mellin-Barnes representation 2. Comparing the series in this formula with the series in the formula 4. Indeed, we again use Theorem 3. Thus we use exactly the same technique of the Mellin integral transform as was employed for the functions y k and arrive at the identity 4.

Putting together 4. The validity of the formulas 4. This time, we are allowed to differentiate the power series term by term using the well-known formula for the derivative of the power law function that directly leads to the formulas 4.

In the equation 4. Then the result of Theorem 4. To show this, we use the same reasoning as in the proof of Theorem 4. The procedure that was employed in this section for analytical treatment of the initial-value problem 4. In particular, our method works for equations in the form. This calculus as well as its applications will be considered elsewhere. The authors acknowledge the support of the Kuwait University for their joint research project No.

Al-Bassam, Yu. Luchko, On generalized fractional calculus and its application to the solution of integro-differential equations. Journal of Fractional Calculus 7 , 69— Search in Google Scholar. Al-Kandari, L. Hanna, Yu. Integral Transforms and Special Functions 30 , — Almeida, D. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Communications in Nonlinear Science and Numerical Simulation 16 , — Dimovski, Operational calculus for a class of differential operators.

Dimovski, Convolutional Calculus , Kluwer Acad. Dzrbashjan, On the integral transformations generated by the generalized Mittag-Leffler function. Nauk Armen. SSR 13 , 21—63 in Russian. Gorenflo, Yu. Luchko, Operational method for solving generalized Abel integral equations of second kind. Integral Transforms and Special Functions 5 , 47— Luchko, F. Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation.

Computational and Applied Mathematics 11 , — Integral Transforms and Special Functions 25 , — Hilfer, Yu. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives.

## On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

For this convolution, we also determine the divisors of zero. Its solution is obtained in terms of the four-parameters Wright function of the second kind. Fractional calculus of variations is one of the recent and important research areas in Fractional Calculus FC. It is a natural extension of the calculus of variations to the case of functionals that depend not only on integer order derivatives, but also on fractional derivatives [ 3 , 23 ]. As in the case of conventional calculus of variations, a necessary condition for a function that solves an optimization problem for a functional depending on some fractional derivatives can be formulated in terms of a suitably modified Euler-Lagrange equation [ 3 ]. The Euler-Lagrange equations of the fractional calculus of variations are very special fractional differential equations that as a rule contain both the left-and right-hand sided fractional derivatives as well as their compositions.

The purpose of this paper is to investigate the existence of solutions to the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivative , , , , where , are Caputo fractional derivatives, , , , and. Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed to guarantees not only the global existence of solutions on the interval , but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to. Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations with -Laplacian on the half-axis follow as a special case of our results.

Mehar Chand 1 , , Jyotindra C. Furthermore, by employing some integral transforms on the resulting formulas, we obtained some more image formulas and also develop a new and further generalized form of the fractional kinetic equation involving the family of some extended generalized Gauss hypergeometric functions and the manifold generality of the family of functions is discussed in terms of the solution of the fractional kinetic equation. The results obtained here are quite general in nature. Singh , Certain fractional kinetic equations involving the product of generalized k-Bessel function, Alexandria Engineering Journal , 55 , Google Scholar. Singh , Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform, Alexandria Engineering Journal , 57 ,

Historical Survey The Modern Approach The Riemann-Liouville Fractional Integral The Riemann-Liouville Fractional Calculus Fractional Differential Equations.

## An Introduction to the Fractional Calculus and Fractional Differential Equations

Elevar la calidad del servicio que la revista presta a los autores. Asegurar la eficacia y la mejora continua del servicio. In this paper we propose a fractional differential equation for the electrical RC and LC circuit in terms of the fractional time derivatives of the Caputo type.

Дэвид. - Это Стратмор, - прозвучал знакомый голос. Сьюзан плюхнулась обратно в ванну.

### Fractional calculus

Никто лучше его не знал, как тщательно следило агентство за своими сотрудниками, поэтому сообщения, приходящие на этот пейджер, как и отправляемые с него, Стратмор старательно оберегал от чужих глаз. Сьюзан опасливо огляделась. Если до этого Хейл не знал, что они идут, то теперь отлично это понял. Стратмор нажал несколько кнопок и, прочитав полученное сообщение, тихо застонал. Из Испании опять пришли плохие новости - не от Дэвида Беккера, а от других, которых он послал в Севилью.

Сьюзан хотелось потянуть шефа назад, в безопасность его кабинета. В кромешной тьме вокруг ей виделись чьи-то лица. На полпути к ТРАНСТЕКСТУ тишина шифровалки нарушилась. Где-то в темноте, казалось, прямо над ними, послышались пронзительные гудки. Стратмор повернулся, и Сьюзан сразу же его потеряла.

Если Танкадо не понял, что стал жертвой убийства, зачем ему было отдавать ключ. - Согласен, - сказал Джабба.  - Этот парень был диссидентом, но диссидентом, сохранившим совесть. Одно дело - заставить нас рассказать про ТРАНСТЕКСТ, и совершенно другое - раскрыть все государственные секреты. Фонтейн не мог в это поверить. - Вы полагаете, что Танкадо хотел остановить червя. Вы думаете, он, умирая, до последний секунды переживал за несчастное АНБ.

Мелькнул лучик надежды. Но уже через минуту парень скривился в гримасе. Он с силой стукнул бутылкой по столу и вцепился в рубашку Беккера. - Она девушка Эдуардо, болван.

Так, значит, вы не по поводу моей колонки. - Нет, сэр. Казалось, старик испытал сильнейшее разочарование. Он медленно откинулся на гору подушек.

Его крик эхом отозвался в черноте, застилавшей. Беккер не знал, сколько времени пролежал, пока над ним вновь не возникли лампы дневного света. Кругом стояла тишина, и эту тишину вдруг нарушил чей-то голос. Кто-то звал. Он попытался оторвать голову от пола.

Думаю, нет нужды спрашивать, куда направился Дэвид, - хмуро сказала .

Рекомендации? - выпалил Джабба.  - Вы ждете рекомендаций. Что ж, пожалуйста. Хватит путаться у нас под ногами, вот моя рекомендация.

- Что показалось тебе странным. Сьюзан восхитилась спектаклем, который на ее глазах разыгрывал коммандер. - ТРАНСТЕКСТ работает с чем-то очень сложным, фильтры никогда ни с чем подобным не сталкивались. Боюсь, что в ТРАНСТЕКСТЕ завелся какой-то неизвестный вирус. - Вирус? - снисходительно хмыкнул Стратмор, - Фил, я высоко ценю твою бдительность, очень высоко.

Джабба нажал на клавишу. И в следующую секунду все присутствующие поняли, что это было ошибкой. ГЛАВА 119 - Червь набирает скорость! - крикнула Соши, склонившаяся у монитора в задней части комнаты.

В вашем номере проститутка. Немец нервно посмотрел на дверь в ванную. Он явно колебался.

Беккер пожал плечами и вгляделся в надпись. - Да, их тут немало. - Я что-то не понимаю, - вмешался Фонтейн.  - Чего мы медлим. - Сэр, - удивленно произнесла Сьюзан, - просто это очень… - Да, да, - поддержал ее Джабба.

Мужчина засмеялся: - Que fea. Ничего себе зрелище.  - Он покачал головой и возобновил работу. Дэвид Беккер стоял в центре пустого зала и думал, что делать. Весь вечер оказался сплошной комедией ошибок.

Так вот почему Дэвид отложил поездку в Стоун-Мэнор. - Сегодня утром я послал за ним машину. Он сказал, что позвонит тебе перед вылетом. Прости, я думал… - Зачем вы послали его в Испанию. Стратмор выдержал паузу и посмотрел ей прямо в .

Она чувствовала себя атеистом, лицом к лицу столкнувшимся с Господом Богом. - Если этот шифр станет общедоступным, - прошептала она, - криптография превратится в мертвую науку. Стратмор кивнул: - Это наименьшая из наших проблем.

- Она улыбнулась и села напротив шефа. Стратмор был крупным кряжистым мужчиной, чье невыразительное лицо скрывало присущие ему решительность, настойчивость и неизменное стремление к совершенству. Серые глаза светились уверенностью, с которой сочеталась профессиональная скрытность, но сегодня в них проглядывали беспокойство и нерешительность. - У вас испуганный вид, - сказала Сьюзан.

- Мой человек ликвидировал его, но не получил ключ. За секунду до смерти Танкадо успел отдать его какому-то туристу. - Это возмутительно! - взорвался Нуматака.