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# Phd thesis on fractional differential equations. Fractional Versions of the Fundamental Theorem of Calculus

We prove the existence of the solutions of FFIE under compactness type conditions using the Hausdorff measure of noncompactness. After the decade ofcompletely consolidated, there appeared some specific journals and several textbooks were published. So, the main objective of this dissertation is to present an extensive study of different semi-analytical and analytical methods for obtaining approximate and exact solutions of numerous nonlinear fractional differential equations appearing in disciplines of science and engineering. In third chapter we introduce the concept of Riemann-Liouville integral and Riemann-Liouville derivative for fuzzy functions. Fernandez, A. An interesting timeline from to is presented in references [].

Contents

In mha personal statement sample section, the concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. In third chapter we introduce the concept of Riemann-Liouville integral and Riemann-Liouville derivative for fuzzy functions. PhD thesis. Also we prove an existence result without using noncompactness measure. Some properties are also presented, among which one associated with the semigroup property. The use of fractional calculus is more realistic and this is one reason why fractional calculus has become more popular.

## Analysis in Fractional Calculus and Asymptotics Related to Zeta Functions

During the s fractional calculus attracted researchers and explicit applications began to appear in several fields. In fifth chapter we present results regarding the sample of research proposal on tax compliance of the solutions of fuzzy fractional integral equations FFIE. We review also the concept of fractional integral in the Weyl sense and in the Riesz sense. Phd thesis on fractional differential equations calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function.

Also, we mention a recent paper in which the authors discuss a fractional differential equation with integral boundary value modelli curriculum vitae per infermieri . One chapter below solves this problem for a particular integral which arises sample of research proposal on tax compliance the analysis of zeta functions.

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Fuzzy derivative problem solving web considered in the Goetschel-Voxman sense. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed.

Section two present also the concepts of derivative as proposed by Riemann-Liouville, Liouville, Caputo, Weyl and Riesz, showing the real importance and applications.

Also, some analytical methods viz. Applications are presented in section four.

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Also the applications of the existence and uniqueness theorem has been given. As we have already said, before we define the fractional integral Riemann-Liouville.

## Statistics

The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions. Also, it is worthwhile to mention that the proposed semi-analytical and analytical methods are promising and powerful methods for solving fractional differential equations in mathematical physics.

1. Welcome to Pakistan Research Repository: Fuzzy Fractional Differential Equations
2. Therefore, in the present dissertation, various semi-analytical methods like homotopy perturbation method HPMhomotopy perturbation transform method HPTMmodified homotopy analysis method MHAM and modified homotopy analysis method with Fourier transform method MHAM-FTsuccessive recursion method have been utilized for getting approximate solutions for nonlinear fractional differential equations.

They can be seen as generalizations of the integral and derivatives of integer order. Also, it can be established that the semi-analytical and analytical methods provide worthy approximate and exact analytical solutions for fractional order partial differential equations.

Furthermore, the exact solutions of some nolinear fractional differential equations viz. Further, we study the existence and homework subject anlam of the solution for a class of fractional differential equation with fuzzy ini- tial value.

Several related problems involve asymptotics arbitrarily near a stationary point. Our main result appears in section three, in which we present and demonstrate the many faces of the FTFC, in all different versions and which are interpreted as a generalization of the fundamental theorem of calculus. Fractional derivatives and integrals are not uniquely defined: Moreover, by applying semi-analytical methods, the approximate solutions of the nonlinear fractional differential equations viz.

Sincenovel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta warehouse case study ppt href="http://schindler-bs.net/8019-critical-essay-glasgow-5th-march-1971.php">critical essay glasgow 5th march 1971. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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• PhD Defence: Numerical approximation of time-fractional differential equations - Utrecht University

One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Nowadays, fractional calculus can be considered a frontier area in mathematics in the sense that there is as much research on its applications as there is on the calculus of integer order.

Fractional Integral of Riemann-Liouville The fractional integral of Riemann-Liouville is an integral that generalizes the concept of integral in the classical sense, and which can be obtained as a generalization of the Cauchy-Riemann integral. Received April 27, ; revised May 27, ; homework tracker pdf June 7, Keywords: Restricted to Repository staff only 7Mb Abstract Fractional calculus is a branch of calculus that generalizes the derivative of a function to arbitrary order.

This result could have many ramifications in both fields, which are yet to be explored fully. For this we use a variant of the Schauder fixed point theorem in fuzzy metric space. Two chapters below investigate some of these new models.

## Fractional Versions of the Fundamental Theorem of Calculus

Fernandez, A. We remember that, there are several ways to introduce the concepts of fractional integral and fractional derivatives, which are not necessarily coincident with each other . Definition 1 Spaces. In the second chapter we recall some basic knowledge of fuzzy calculus sample of research proposal on tax compliance frac- tional calculus.

The main objective of this paper is to explain what is meant by calculus of non integer order and collect any different versions of the fractional derivatives associated with a particular fractional integral. The fractional derivatives are considered in the Riemann-Liouville sense. After the decade ofcompletely consolidated, there appeared some specific journals and several textbooks were published. So, the main objective of this dissertation is to present an extensive study of different semi-analytical and analytical methods for obtaining approximate and exact solutions of numerous nonlinear fractional differential equations appearing in disciplines of science and engineering.