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This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system's state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed. [ABSTRACT FROM AUTHOR]

This paper studies distributed optimal control for non-cooperative systems involving time-fractional hyperbolic operators. Through the application of the Lax–Milgram theorem, we confirm the existence and uniqueness of weak solutions. Central to our approach is the utilization of the linear quadratic cost functional, which is meticulously crafted to encapsulate the interplay between the system’s state and control variables. This functional serves as a pivotal tool in imposing constraints on the dynamic system under consideration, facilitating a nuanced understanding of its controllability. Using the Euler–Lagrange first-order optimality conditions with an adjoint problem defined by means of the right-time fractional derivative in the Caputo sense, we obtain an optimality system for the optimal control. Finally, some examples are analyzed.

This paper is concerned with the study of a system of q-fractional differential equations. By applying the Leray-Schauder alternative, we show the existence of multiple solutions to our system. Then, the uniqueness of the solution is shown using the Banach fixed point theorem. On the other hand, we study stability in the sense of Ulam-Hyers then that of Ulam-Hyers Rassias. Finally, we give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Vallis proposed a simple model for El-Niño weather phenomenon (referred as Vallis system) by adding an additional parameter p to the Lorenz system. He showed that the chaotic behavior of the Vallis system is related to the El-Niño effect. In the present article we study fractional version of Vallis system in detail. We investigate bifurcations and chaos present in the fractional Vallis system and the effect of variation of system parameter p. It is observed that the range of values of parameter p for which the Vallis system is chaotic, reduces with the reduction of the fractional order. Further we analyze the incommensurate fractional Vallis system and find the critical value below which the system loses chaos. We also synchronize Vallis system with Bhalekar-Gejji system. [ABSTRACT FROM AUTHOR]

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme.

Many real-life phenomena in physics, engineering, biology, medicine, economics, etc. can be modeled by fractional delay differential equations and having in mind that these modeling interpret phenomena better, they are noticed by researchers, engineers, and mathematicians. In this paper the method of fractional finite differences has been presented and used for numerical solution of such models and used for solving a number of famous fractional order version of models such as the fractional order version of Hutchinson model is related to rate of population growth, the fractional order version of Verhulst Pearl model is related to the impact of a specific factor on the changes of population in an area model, fractional order version of the negative impact of population growth in a specific time and model of fractional order version of the four years life cycle of a population of lemmings. The proposed method besides being simple is so exact which is sensible in the solved problems.

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann−Liouville and Caputo’s derivatives converge, on long times, to the Grünwald−Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications.

Abstract: Many real-life phenomena in physics, engineering, biology, medicine, economics, etc. can be modeled by fractional delay differential equations and having in mind that these modeling interpret phenomena better, they are noticed by researchers, engineers, and mathematicians. In this paper the method of fractional finite differences has been presented and used for numerical solution of such models and used for solving a number of famous fractional order version of models such as the fractional order version of Hutchinson model is related to rate of population growth, the fractional order version of Verhulst Pearl model is related to the impact of a specific factor on the changes of population in an area model, fractional order version of the negative impact of population growth in a specific time and model of fractional order version of the four years life cycle of a population of lemmings. The proposed method besides being simple is so exact which is sensible in the solved problems. [Copyright &y& Elsevier]

Abstract: In this paper, we present a powerful modification of the rectangular decomposition method that is used to solve fractional diffusion-wave equations. The new modification extends the application of the rectangular decomposition method to solve a wide class of space–time fractional partial differential equations. The proposed modification gives an analytical solution in the form of a convergent series with easily computable components. Some numerical examples are given to illustrate the efficiency and reliability of the method. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, a new modification of the Adomian decomposition method is effectively implemented for solving fractional diffusion-wave equations that will facilitate the calculations, where the fractional derivative is based on Caputo definition. The proposed algorithm gives an analytical solution in the form of a convergent series with easily computable components. Numerical examples are given to show the application of the present algorithm. The new modification introduces a promising tool for many partial differential equations. [Copyright &y& Elsevier]

In the present article, we implement a new numerical scheme, the quasilinearized semi-orthogonal B-spline wavelet method, combining the semi-orthogonal B-spline wavelet collocation method with the quasilinearization method, for a class of multi-term non-linear fractional order equations that contain both the Riemann–Liouville fractional integral operator and the Caputo fractional differential operator. The quasilinearization method is utilized to convert the multi-term non-linear fractional order equation into a multi-term linear fractional order equation which, subsequently, is solved by means of semi-orthogonal B-spline wavelets. Herein, we investigate the operational matrix and the convergence of the proposed scheme. Several numerical results are delivered to confirm the accuracy and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

Several fractional-order operators are available and an in-depth knowledge of the selected operator is necessary for the evaluation of fractional integrals and derivatives of even simple functions. In this paper, we reviewed some of the most commonly used operators and illustrated two approaches to generalize integer-order derivatives to fractional order; the aim was to provide a tool for a full understanding of the specific features of each fractional derivative and to better highlight their differences. We hence provided a guide to the evaluation of fractional integrals and derivatives of some elementary functions and studied the action of different derivatives on the same function. In particular, we observed how Riemann–Liouville and Caputo's derivatives converge, on long times, to the Grünwald–Letnikov derivative which appears as an ideal generalization of standard integer-order derivatives although not always useful for practical applications. [ABSTRACT FROM AUTHOR]