Solution Of Optimality System Research Articles

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation [Formula: see text] is studied by the symmetry analysis method. Liu et al. obtained two group generators [Formula: see text] [Formula: see text] and only one trivial solution [Formula: see text] for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case [Formula: see text], we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

Lie symmetry analysis, optimal system and exact solutions for a NLPDE from the reduced quasi-classical self-dual Yang–Mills equation

Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

In this paper, the (2+1)-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of (1+1)-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth

Abstract In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.

Parametric Study and Optimization of Hydrogen Production Systems Based on Solar/Wind Hybrid Renewable Energies: A Case Study in Kuqa, China

Based on the concept of sustainable development, to promote the development and application of renewable energy and enhance the capacity of renewable energy consumption, this paper studies the design and optimization of renewable energy hydrogen production systems. For this paper, six different scenarios for grid-connected and off-grid renewable energy hydrogen production systems were designed and analyzed economically and technically, and the optimal grid-connected and off-grid systems were selected. Subsequently, the optimal system solution was optimized by analyzing the impact of the load data and component capacity on the grid dependency of the grid-connected hydrogen production system and the excess power rate of the off-grid hydrogen production system. Based on the simulation results, the most matched load data and component capacity of different systems after optimization were determined. The grid-supplied power of the optimized grid-connected hydrogen production system decreased by 3347 kWh, and the excess power rate of the off-grid hydrogen production system decreased from 38.6% to 10.3%, resulting in a significant improvement in the technical and economic performance of the system.

On the Optimal System and Series Solutions of Fifth-Order Fujimoto-Watanabe Equations

This paper investigates the two fifth-order Fujimoto-Watanabe equations from the perspective of the group theoretic approach. We identify the reduced equations that lead to the solutions of these high order equations. Furthermore, the corresponding solutions are found by power series due to their nonlinear characteristics. As a result, the findings of the study demonstrate the convergence of solutions for such models and identifies the travelling wave solutions.

The Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional variable coefficients integrable coupled Burgers equations

The variable coefficients integrable coupled Burgers equations (vcICBs) are researched by the Lie symmetry analysis method. The infinitesimal generators, commutative relations and adjoint relations are given. Optimal system of vcICBs are studied by Olver’s method. Based on the optimal system, the (2+1)-dimensional vcICB equations are reduced to (1+1)-dimensional equations. Some new exact solutions of vcICB equations are derived. Some kink solutions, breathe wave solutions and soliton solutions are got by utilising -expansion method, Riccati equation method, Tanh-expansion method and Painlevé analysis method. By altering different coefficient functions, many different types of exact solutions can be derived. The effect of the coefficient functions on the solution is analysed. By studying the evolution of the solution with time t, the dynamic behaviors of the solutions are analysed.

Symmetry analysis, optimal system, and invariant solutions for a (2+1)-dimensional two-phase mass flow model

The paper investigates a two-phase mass flow model governed by gravity, involving solid particles and a viscous fluid. By utilizing the Lie symmetries admitted by the system, similarity solutions for the (2+1)-dimensional two-phase mass flow model are obtained. A comprehensive set of local point symmetries is established, and a well-suited collection of two-dimensional subalgebras is constructed from the maximal Lie invariance algebra. The optimal system’s vector fields are then utilized to directly reduce the governing model to a system of ordinary differential equations. Through analytical solutions, we successfully solve the resulting systems and further analyze their physical behaviors numerically.

Lie symmetry analysis, optimal system, symmetry reductions and analytic solutions for a (2＋1)-dimensional generalized nonlinear evolution system in a fluid or a plasma

Nonlinear evolution equations are used to describe such nonlinear phenomena as the solitons, travelling waves and breathers in fluid mechanics, plasma physics and optics. In this paper, we investigate a (2＋1)-dimensional generalized nonlinear evolution system in a fluid or a plasma. Via the Lie symmetry analysis, we acquire the Lie point symmetry generators and Lie symmetry groups of that system. Via the optimal system method, we derive the optimal system of the 1-dimensional subalgebras. Based on the symmetry generators in that optimal system, we give some symmetry reductions for the (2＋1)-dimensional generalized nonlinear evolution system. Finally, via those symmetry reductions, we acquire some soliton, rational-type and power-series solutions.

Switching decisions in a hybrid MTS/MTO production system comprising multiple machines considering setup

A hybrid make-to-stock (MTS) and make-to-order (MTO) production system was developed and analyzed in this study using a Markov decision process (MDP) model. Previous research successfully modeled cases with and without consideration of setup to obtain optimal solutions for such systems; however, the analyzed models were limited to a single machine. Therefore, this study formulated a hybrid MTS/MTO production system comprising multiple machines using a discrete-time MDP without prioritizing either MTS or MTO. All machines were considered hybrid machines that could be switched between MTS and MTO production by incurring a setup period. The MDP model was formulated using the state space, action space of the switching decision, transition probability, and cost function. The objective was to minimize the average cost per period, which comprised the MTS inventory holding costs, MTO job backlog costs, setup costs, and lost sales costs. A series of numerical investigations was undertaken to examine the number of machines that should be switched simultaneously within a given period and evaluate the dedication of different numbers of machines to a particular process, which are unique conditions to the assumption of multiple machines. The results indicated that switching only one machine at a time in a given period was sufficient and that the average cost remained unchanged even when machines were partially dedicated to MTS and/or MTO production. Thus, the proposed switching decisions were shown to realize the efficient control of hybrid MTS/MTO production systems.

Experimental assessment of an unknown-input estimator for a nonlinear wave energy converter

In the wave energy field, one of the main challenges towards commercialisation of wave energy devices is the development of suitable control laws, able to maximise the absorbed energy while guaranteeing effective satisfaction of any required physical constraint. However, one of the main characteristics of this optimal control problem is that the system behaviour is strongly influenced by the external (uncontrollable) input arising from the wave source, i.e. the wave excitation, which is often unmeasurable. As such, computation of optimal control solutions for WEC systems requires availability of instantaneous knowledge of the wave excitation, and hence input-unknown estimators are developed within the control loop. State-of-the-art estimation strategies are based on the knowledge of control-oriented linearized models of the system, often neglecting the influence of nonlinear phenomena within the system description. We propose, in this paper, an approach inspired by disturbance observer-based control, able to accommodate well-known hydrodynamic nonlinear effects in the process of estimating the unknown excitation force acting on the device. This strategy, which in contrast to the usually applied estimators does not require an implicit/explicit model of the wave excitation force, is tested on a hardware-in-the-loop facility in different sea state conditions, to realistically assess its performance in terms of estimation error and delay. The experimental appraisal show satisfactory results in terms of normalized root mean square error and average delay, which, together with the simplicity of the method, positions the proposed strategy as a promising candidate for hardware implementations in real environments.

Lie symmetry analysis, optimal system and exact solutions of variable-coefficients Sakovich equation

In this paper, the Sakovich equation is extended for the first time to a new equation with variable-coefficients on the time variable. The infinitesimal generators are obtained by performing a Lie symmetry analysis of the equation. Following this, the optimal system of one-dimensional subalgebras of the equation is analyzed. The (2+1)-dimensional variable-coefficients Sakovich equation is then reduced to (1+1)-dimensional partial differential equations by similarity reductions. The reduced partial differential equations are simplified to ordinary differential equations by the traveling wave transform. The new periodic wave solution and two-soliton interaction solution are obtained, and some exact solutions are obtained using the extended tanh-function method and the extended sech-function method. Finally, different types of soliton solutions are obtained by assigning different functions to the coefficient functions, and 3-dimensional figures are used to visualize the structural features of the soliton solutions.

Exact Solutions for the KMM System in (2+1)-Dimensions and Its Fractional Form with Beta-Derivative

Fractional calculus is useful in studying physical phenomena with memory effects. In this paper, the fractional KMM (FKMM) system with beta-derivative in (2+1)-dimensions was studied for the first time. It can model short-wave propagation in saturated ferromagnetic materials, which has many applications in the high-tech world, especially in microwave devices. Using the properties of beta-derivatives and a proper transformation, the FKMM system was initially changed into the KMM system, which is a (2+1)-dimensional generalization of the sine-Gordon equation. Lie symmetry analysis and the optimal system for the KMM system were investigated. Using the optimal system, we obtained eight (1+1)-dimensional reduction equations. Based on the reduction equations, new soliton solutions, oblique analytical solutions, rational function solutions and power series solutions for the KMM system and FKMM system were derived. Using the properties of beta-derivatives and another transformation, the FKMM system was changed into a system of ordinary differential equations. Based on the obtained system of ordinary differential equations, Jacobi elliptic function solutions and solitary wave solutions for the FKMM system were derived. For the KMM system, the results about Lie symmetries, optimal system, reduction equations, and oblique traveling wave solutions are new, since Lie symmetry analysis method has not been applied to such a system before. For the FKMM system, all of the exact solutions are new. The main novelty of the paper lies in the fact that beta-derivatives have been used to change fractional differential equations into classical differential equations. The technique can also be extended to other fractional differential equations.

RETRACTED: Lie symmetry analysis, optimal system and exact solutions for variable-coefficients (2 + 1)-dimensional dissipative long-wave system

In this article, the symmetry, optimal system, exact solutions and conservation laws of (2+1)-dimensional dissipative long-wave system with variable-coefficients (vcDLW) are investigated. These are important for mathematicians and physicists to solve problems about shallow water waves. The infinitesimal generators of independent variables are solved using Lie symmetry analysis method firstly. Based on the linear combination of vector fields, the representative elements of one-dimensional subalgebras optimal system are introduced by Olver's method. Some new exact solutions of the (2+1)-dimensional vcDLW are obtained by using Riccati equation method and tanh expansion method, such as kink solutions and soliton solutions. It is shown that this system has different solutions for different reductions. These solutions are important for describing a number of physical phenomena. The difference in this article is that the coefficients of some terms are functions. The unique feature of this article is that we can obtain many different types of equations when the coefficient functions change. This makes the (2+1)-dimensional vcDLW describe more general physical phenomena. In addition, the dynamical behaviour of the exact solutions of the (2+1)-dimensional vcDLW is analysed. And the evolution of several types of solutions is studied depending on the change in time. This process can be described better with the help of figures. Finally, The conservation laws of the (2+1)-dimensional vcDLW are obtained using the new conservation theorem with two modificatory rules.

A (2+1)-dimensional modified dispersive water-wave (MDWW) system: Lie symmetry analysis, optimal system and invariant solutions

In this article, the authors study a (2+1)-dimensional MDWW system, which describes the non-linear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The Lie group theoretic approach is employed to find the similarity reductions and analytic solutions of the (2+1)-dimensional MDWW system. The infinitesimal generators for the considered system are obtained under the invariance property of the Lie group of transformations. Later, we construct groups of symmetries and tables for commutation and adjoints. The adjoint table is further used to establish a one-dimensional optimal system of subalgebras. Finally, based on the optimal system, similarity reductions are obtained. A repeated process of similarity reductions reduces the governing system of partial differential equations (PDEs) into systems of ordinary differential equations (ODEs) that generate invariant solutions. Moreover, the dynamical behaviors of the obtained solutions such as multi-soliton, doubly soliton, single soliton, solitary waves, and stationary waves are graphically shown using 3D, 2D, and corresponding contour plots. Thus, physicists and mathematicians can follow complicated physical phenomena more effectively and efficiently using these graphs.

A (2 + 1)-dimensional variable-coefficients extension of the Date–Jimbo–Kashiwara–Miwa equation: Lie symmetry analysis, optimal system and exact solutions

Abstract In this article, the Date–Jimbo–Kashiwara–Miwa equation is extended to a new variable-coefficients equation with respect to the time variable. The infinitesimal generators are acquired by studying the Lie symmetry analysis of the equation, and the optimal system of this equation is presented. After that, the equation performed similarity reductions, and the reduced partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) with the help of traveling wave transform. Then, the exact solutions are found by applying the extended tanh-function method. Finally, the structural features of exact solutions to different times are shown with the help of images.

Symmetries, optimal system, exact and soliton solutions of ([formula omitted])-dimensional Gardner-KP equation

In this research article, the (3+1)-dimensional nonlinear Gardner–Kadomtsov–Petviashvili (Gardner-KP) equation which depicts the nonlinear modulation of periodic waves, is analyzed through the Lie group-theoretic technique. Considering the Lie invariance condition, we find the symmetry generators. The proposed model yields eight-dimensional Lie algebra. Moreover, an optimal system of sub-algebras is computed, and similarity reductions are made. The considered nonlinear partial differential equation is reduced into ordinary differential equations (ODEs) by utilizing the similarity transformation method (STM), which has the benefit of yielding a large number of accurate traveling wave solutions. These ODEs are further solved to get closed-form solutions of the Gardner-KP equation in some cases, while in other cases, we use the new auxiliary equation method to get its soliton solutions. The evolution profiles of the obtained solutions are examined graphically under the appropriate selection of arbitrary parameters.

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Lie symmetry analysis, optimal system and exact solutions of a new (2 + 1)-dimensional KdV equation

This work attempts to apply the Lie symmetry approach to an updated (2+1)-dimensional KdV equation, recently updated in A.-M. Wazwaz, Nucl. Phys. B 954 (2020) 115009. The equation can be considered as one of the famous examples of the soliton equation. The infinitesimal generators for the governing equation have been found using the invariance property of Lie groups. The commutator table, adjoint table, invariant functions and one-dimensional optimal system of subalgebras are then derived using Lie point symmetries. Some group invariant solutions are derived based on various subalgebras, symmetry reductions and an optimal system. To demonstrate the physical acceptability of the results, the obtained solutions are evaluated using numerical simulation.