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Applications of Fractional Processes
Abstract
Chapter 2 described how fractional calculus can be applied to generate fat-tailed distributions; discussed how to apply fractional processes to the pricing of derivatives. As fractional processes are not semi martingales, violations of the no-arbitrage condition might occur. We have seen how to circumvent this problem.
We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.
In this work, we propose an implicit finite-difference scheme to approximate the solutions of a generalization of the well-known Klein-Gordon-Zakharov system. More precisely, the system considered in this work is an extension to the spatially fractional case of the classical Klein-Gordon-Zakharov model, considering two different orders of differentiation and fractional derivatives of the Riesz type. The numerical model proposed in this work considers fractional-order centered differences to approximate the spatial fractional derivatives. The energy associated to this discrete system is a non-negative invariant, in agreement with the properties of the continuous fractional model. We establish rigorously the existence of solutions using fixed-point arguments and complex matrix properties. To that end, we use the fact that the two difference equations of the discretization are decoupled, which means that the computational implementation is easier that for other numerical models available in the literature. We prove that the method has square consistency in both time and space. In addition, we prove rigorously the stability and the quadratic convergence of the numerical model. As a corollary of stability, we are able to prove the uniqueness of numerical solutions. Finally, we provide some illustrative simulations with a computer implementation of our scheme.
UDC 517.9We develop the existence theory for a more general class of nonlocal integro-multipoint boundary value problems ofCaputo type fractional integro-differential inclusions. Our results include the convex and non-convex cases for the givenproblem and rely on standard fixed point theorems for multivalued maps. The obtained results are illustrated with the aidof examples. The paper concludes with some interesting observations.
In this paper, we discuss the existence of solutions for generalized Caputo fractional differential equations and inclusions equipped with Steiltjestype fractional integral boundary conditions via fixed-point theory. Examples are constructed for illustrating the obtained results.
We present the criteria for the existence of solutions for a nonlinear mixed-order coupled fractional differential system equipped with a new set of integral boundary conditions on an arbitrary domain. The modern tools of the fixed point theory are employed to obtain the desired results, which are well-illustrated by numerical examples. A variant problem dealing with the case of nonlinearities depending on the cross-variables (unknown functions) is also briefly described. © 2021 B. Ahmad, S. Hamdan, A. Alsaedi and S.K. Ntouyas, licensee AIMS Press.
This paper deals with the existence and uniqueness of solution for the Cauchy problem of φ−Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.
The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory.
In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.
We investigate the existence of solutions for integro-multipoint boundary value problems involving nonlinear multi-term fractional integro-differential equations. The case involving three different types of nonlinearities is also briefly described. The desired results are obtained by applying the methods of modern functional analysis and are well-illustrated with examples.
A numerical formula is derived which gives solutions of the fractional Schrödinger equation in time-independent form in the case of Coulomb potential using Riemann–Liouville definition of the fractional derivative and the quadrature methods. The formula is applied for electron in the nucleus field for multiple values of fractional parameter of the space-dependent fractional Schrödinger equation and for each value of the space-dependent fractional parameter, multiple values of energies are applied. Distances are found at which the probability takes its maximum value. Values of energy obtained in this study corresponding to the maximum value of probability are compared with the energy values resulted from the fractional Bohr’s atom formula in the fractional quantum mechanics.
A generalization of the economic order quantity (EOQ) model is taken into account to study the memory effect in the inventory model. Fractional calculus is one of the efficient medium to establish the existence of memory effect in the economic system. This paper deals with a system of two fractional order differential equations to govern the fractional order EOQ model with partial backlogging. The fractional order differential equation is considered here in Caputo sense. The theory of memory kernel is applied to set up the memory dependent EOQ model. Using primal geometric programming method, analytical solution of the proposed problem is found. Using empirical values of the system parameters, we have established the concept of short and long memory effect. It is clear from the investigations that the strength of memory can be controlled by the order of fractional derivative or integration. Sensitivity analysis has been carried out for long memory and short memory affected problem both to identify the important model parameters for different situations. Finally, the manuscript is concluded with some recommendation about the memory dependent EOQ model.
A comment on an article by Khater et al published in Pramana – J. Phys. 90: 59 (2018) is presented here. We represent two quotes on the article, the two quotes about the space-dependent fractional Schrödinger equation type and about one of the constants used by the authors.
In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials , and allows to transform fractional order systems in FOPDE that only contains integer-order derivatives. As a consequence of considering the extra dimension, the geometric interpretation of fractional derivatives and integrals naturally emerges as the area under the curve of a characteristic trajectory and as the direction of a tangent characteristic vector, respectively. Besides this, a new physical interpretation is suggested for the fractional derivatives, integrals and dynamical systems. MSC 2010 : 26A33, 34A08, 35F9, 45D05
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