Open Access
Articles
Article ID: 2447
by Haihua Wang, Jie Zhao, Junhua Ku, Yanqiong Liu
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 19 Views,
In the context of solving $(k, \Psi)$-Hilfer fractional differential equations with Sobolev type, we initially explore a more generalized version of the $(\alpha, \beta, k)$-resolvent family. Subsequently, we present various properties associated with this resolvent family. Specific instances of this resolvent family, such as the $C_0$ semigroup, sine family, cosine family and others, have been previously discussed in other academic papers. By combining this resolvent family with the $(k, \Psi)$-Hilfer fractional derivative, we examine the existence of mild solutions to $(k, \Psi)$-Hilfer Sobolev type fractional evolution equations, without requiring the existence of the inverse of $E$. Ultimately, two existence theorems are derived.
Open Access
Articles
Article ID: 2448
by T. D.Omurov, K. R.Dzhumagulov
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 20 Views,
In this paper, we study the inverse problem for the wave equation with the second-order d’Alembert operator in an unbounded domain in a space with a non-uniform metric. For physical applications, inverse problems for second-order partial differential equations are of particular interest. Such inverse problems are encountered in the study of wave processes, processes of electromagnetic interactions, as well as in various reduction processes. Moreover, if there are external acting forces with respect to the indicated equations that allow additional information about the solution of the original equations, then we obtain classes of inverse problems of a coefficient nature with the d’Alembert operator, which are of particular interest to scientists in this field, in which the results of this article are relevant. Also, the relevance of the problem under study is due to the fact that it is an inverse problem, where the sought quantities are the causes of some known consequences of a particular process. Whereas for direct problems, the methods for solving are well known. Thus, this paper provides a solution to the inverse problem of mathematical physics with a hyperbolic operator and generalizes existing results.
Open Access
Articles
Article ID: 2449
by Savita Garg, Neetu Rani
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 50 Views,
This paper presents a control process of reliability models for key subsystems in a coil shop by advanced differential equations approach. The goal is to optimize the maintenance scheduling process for the critical subsystems, enabling the system to operate at maximum efficiency. Maintenance strategies significantly influence this outcome, and selecting the right strategy is not trivial. The designed control process model can be applied by administrative setup in manufacturing concerns. The 3-state models here are developed for the dynamic behavior of the system under the impact of preventive maintenance strategies. Both maintenance and repair of the units are perfect. The numerical analysis of the system is also discussed to compare the behavior of the present model and the proposed models. The comparison helps in findings the production-affecting factors and addresses maintenance planning gaps for critical subsystems. This approach aims to optimize the entire manufacturing system, potentially increasing profitability.
Open Access
Articles
Article ID: 2450
by Indrajeet Varhadpande, V. Murthy, N. K.Lamba
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 16 Views,
Recent advances in the understanding of the precise physical thermal behaviour of various solids under the effect of fractional-order derivatives have boosted the study of thermoelasticity, which is primarily important in various industrial designs of usable structural materials. We investigated a thin circular plate that was subjected to additional sectional heating on its top and lower surfaces while creating thermal insulation around its outside border. In this work, we maintained the heat transfer equation while accounting for the impact of Caputo-Fabrizio fraction-order derivatives. According to specified boundary constraints, the integral transformation approach is used to assess the analytical solution of the displacement, temperature change, and thermal stresses. Furthermore, various functions and fractional parameters are computed using the material properties of aluminium metal plates for numerical purposes.
Open Access
Articles
Article ID: 2451
by Asrar SalehAlsulami, Mariam Al-Mazmumy, Maryam AhmedAlyami, Mona Alsulami
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 17 Views,
In this article, we generalize the fractional Riccati differential equations (FRDEs) by using a fractional derivative of a function with respect to another function ( $\psi$-Caputo derivative) and obtain $\psi$-FRDEs. Using the Adomian decomposition method (ADM) with Wazwaz modification, we solve the $\psi$-FRDEs semi-analytically. Comparing the solutions of the $\psi$-FRDEs with several functions of $\psi(x)$ and different values of fractional orders, we show that the presented method is efficient.
Open Access
Articles
Article ID: 2452
by Wan, Suliadi FirdausSufahani, Mahmod Mohamad, Norhaslinda ZainalAbidin
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 55 Views,
This research delves into the application of the modified shooting method for the numerical resolution of non-standard optimal control (OC) problems. More precisely, it concentrates on scenarios where the final state value component remains unknown and unconstrained, leading to a non-zero final shadow value or costate variable. Moreover, the objective function involved a piecewise royalty function, which poses a challenge due to its lack of differentiability within a specific time interval. Consequently, the novel modified shooting method was employed to ascertain the elusive final state value. The model’s differentiability is maintained throughout by adopting a continuous hyperbolic tangent (tanh) approximation. In addition, the construction of the Sufahani-Ahmad-Newton-Golden-Royalty Algorithm (SANGRA) and Sufahani-Ahmad-Powell-Golden-Royalty Algorithm (SAPGRA) was accomplished using the C++ programming language to formulate the problem. The outcomes of these algorithms, satisfying the criteria for optimality, were juxtaposed with non-linear programming (NLP) techniques such as Euler and Runge-Kutta, aside from Trapezoidal and Hermite-Simpson approximations. This groundbreaking discovery carries extensive practical implications as it propels the field forward and ensures the application of contemporary problem-solving methodologies. Moreover, the study underscores the significance of fundamental theory in effectively tackling current economic challenges.
Open Access
Articles
Article ID: 2453
by Mohamed H.Hendy, Magdy A.Ezzat, Esraa M.Al-lobani, Ahmed S.Hassan
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 52 Views,
A new study has been developed that considers the size-dependent interaction between viscoelastic deformation and thermal fields, incorporating the fractional heat conduction law with and without energy dissipation. The model is used for a particular one-dimensional problem involving a polymer micro-rod of arbitrary length experiencing three different types of thermal loading without the presence of any heat source. The study uses Laplace transforms and numerical inversion to examine how fractional order, nonlocal elasticity, and nonlocal thermal conduction impact thermal dispersion and thermo-viscoelastic response. Comparative numbers demonstrate the effects of various parameters. Findings demonstrate that nonlocal thermal and viscoelastic characteristics have a significant impact on all recorded field values, offering possible suggestions for the creation and assessment of thermal-mechanical attributes in nanoscale devices.
Open Access
Articles
Article ID: 2454
by Yaya SAGNA, Lamine SYLLA, Sadibou AIDARA
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 23 Views,
This paper deals with a class of generalized backward stochastic differential equations driven by two mutually independent fractional Brownian motions (FGBSDEs in short). The existence and uniqueness of solutions for FGBSDE as well as a comparison theorem are obtained.
Open Access
Articles
Article ID: 2455
by Mona Magzoub, Tarig M.Elzaki, Mourad Chamekh
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 21 Views,
This work investigates and solves the time-fractional telegraph equations (TFTEs) occurring in electromagnetism, which serve as mathematical models in several practically significant applied research domains. Elzaki transform (ET) is employed in this process. Caputo sense describes fractional derivatives. Solutions of TFTEs were found in an easy-to-understand, step-by-step way using ET. In addition, instances are given to show how the phrase can be applied and how valid it is for the problem-solving form. The exact solutions and the analytical solutions accord well for the tested problems. This work also discusses the convergence of the ET technique to the exact solution of TFTEs. Several examples of linear and nonlinear TFTEs are used to demonstrate the suggested methodology. The novel technique’s results show that it is an effective way to solve TFTEs, and it makes the procedure easier.
Open Access
Articles
Article ID: 2456
by Aysel T.Ramazanova
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 27 Views,
We consider a nonsmooth optimal control problem described by a system of second-order hyperbolic equations with Goursat boundary conditions. A number of necessary optimality conditions are proved in terms of directional derivatives.
Open Access
Articles
Article ID: 2477
by Wan Noor Afifah Wan Ahmad, Suliadi Firdaus Sufahani, Mahmod Abd Hakim Mohamad, Razamin Ramli
Advances in Differential Equations and Control Processes, Vol.31, No.4, 2024; 357 Views,
An introduction to optimal control, a fundamental concept in engineering and science disciplines, is a process of finding ways of controlling dynamic systems in such a way that they achieve certain goals while being subjected to given state limitations. The conventional approaches developed in the past have been integral to solving optimal control problems, including the maximum principle belonging to Pontryagin and the dynamic programming method. While relatively straightforward, these methods are not always amenable to higher-dimensional scenarios, interacting forces, or other non-trivial constraints. This paper presents a new methodology to extend classical optimal control, considering both direct and indirect optimization techniques. The direct methods, Euler and Runge-Kutta, Trapezoidal, and Hermite-Simpson, do not require the explicit derivation of the analytical control laws to execute control trajectories. Semi-analytical techniques, such as the shooting method, are based the control laws on proposed adjoint differential equations. This paper presents the basic outline of the classical optimal control problem and discusses the direct and indirect optimization methods. It is illustrated by referencing various examples, like the fixed-rate royalty payment approach. After outlining each framework, we identify the positive and negative aspects of the approach in questions of consistency and performance. Finally, the problems on the synergy of direct and indirect methods are considered further, and areas of further development of the presented methods and their integration with the more sophisticated tools, such as machine learning are identified. It can be concluded that the approach of using both direct and indirect optimization methods presents great potential in modernizing the classical optimal control, which challenges the conventional techniques and indicates potential for further development of the control system optimization.
Prof. Dr. Ji-Huan He
Soochow University, China
Excellence in Research for Australia (ERA Journal ID: 41584)
Elektronische Zeitschriftenbibliothek (EDB-Number: 2424167-2)
Eurasian Scientific Journal Index
Scilit Database (Switzerland)
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