Fifth-order Equation Research Articles

Hyers-Ulam Stability of Fifth Order Linear Differential Equations

In this paper, we study the Hyers-Ulam stability for the fifth-order linear differential equation. In particular, we treat $\varsigma$ as an arrangement of differential equation and in the form%\begin{equation*}$$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$$%\end{equation*}where $\varsigma \in c^{5} [k, l]$, $\Omega \in [k, l]$. We demonstrate that$\varsigma^{v}(x)+\eta_1\varsigma^{iv}(x)+\eta_2\varsigma^{'''}(x)+\eta_3\varsigma^{''}(x)+\eta_4\varsigma^{'}(x)+\eta_5\varsigma(x)=\Omega(x)$ has the Hyers-Ulam stability. Two illustrative examples are given to represent the effectiveness of the proposed method. Fifth-order linear differential equations find applications in a wide range of fields, from engineering and control theory to physics, biology, and beyond. These equations are powerful tools for modeling systems with complex dynamics that involve multiple interacting forces or rates of change. Understanding and analyzing their stability and behavior can lead to significant advancements in the design, control, and optimization of these systems.

Modeling of Liquefied Natural Gas Cold Power Generation for Access to the Distribution Grid

Cold energy generation is an important part of liquefied natural gas (LNG) cold energy cascade utilization, and existing studies lack a specific descriptive model for LNG cold energy transmission to the AC subgrid. Therefore, this paper proposes a descriptive model for the grid-connected process of cold energy generation at LNG stations. First, the expansion kinetic energy transfer of the intermediate work mass is derived and analyzed in the LNG unipolar Rankine cycle structure, the mathematical relationship between the turbine output mechanical power and the variation in the work mass flow rate and pressure is established, and the variations in the LNG heat exchanger temperature difference, seawater flow rate, and the turbine temperature difference in the cycle system are investigated. Secondly, based on the fifth-order equation of state of the synchronous generator, the expressions of its electromagnetic power, output AC frequency, and voltage were analyzed. Finally, the average equivalent models of the machine-side and grid-side converters are established using a direct-fed grid-connected structure, thus forming a descriptive model of the overall drive process. The ORC model is built in Aspen HYSIS to obtain the time series expression of the torque output of the turbine; based on the ORC output torque, the permanent magnet synchronous generator (PMGSG) as well as the direct-fed grid-connected structure are built in MATLAB/Simulink, and the active power and current outputs of the grid-following-type voltage vector control method and the grid-forming-type power-angle synchronous control method are also verified.

Oscillatory and Asymptotic Criteria for a Fifth-Order Fractional Difference Equation

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms.

A bidirectional fifth-order partial differential equation: Lax representation and soliton solutions

In this paper, we study the bidirectional fifth-order partial differential equation uttt−uxxxxt−4(uxut)xx−4(uxuxt)x=0, which was proposed as a new integrable equation by Wazwaz [Phys. Scr. 83, 015012 (2011)]. By means of the prolongation structure method of Wahlquist and Estabrook [J. Math. Phys. 16, 1–7 (1975)], we construct a Lax representation for this equation. This enables us to confirm its integrability and identify it as the potential second-order flow in a Gelfand-Dickey hierarchy. We also use the Darboux transformation and present the multi-soliton solutions in the Wronskian form. Finally, we comment on a more general bidirectional partial differential equation.

Frequency studies of nonlinear TSDT FGM plates in the linear shear corrections and thermal environment temperature

The effects of third-order shear deformation theory (TSDT) and varied linear shear correction coefficient on the vibration frequency of thick functionally graded material (FGM) plates with fully homogeneous equations under thermal environment and power law are investigated. The nonlinear coefficient c_1 term of displacement field of TSDT is included in the fully homogeneous equation under vibration of FGM plates. The determinant of the coefficient matrix in dynamic equilibrium differential equations under vibration can be represented in the fully fifth-order polynomial equation, thus the natural frequency can be found. The natural frequency with/without the nonlinear c_1 term of displacement fields is investigated. It is a significant novelty with the consideration of the nonlinear c_1 term in the frequency computation.

On examining the predictive capabilities of two variants of the PINN in validating localized wave solutions in the generalized nonlinear Schrödinger equation

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

A numerical solver based on Haar wavelet to find the solution of fifth-order differential equations having simple, two-point and two-point integral conditions

Modified simple equation technique for first-extended fifth-order nonlinear equation, medium equal width equation and Caudrey–Dodd–Gibbon equation

Modified simple equation technique for first-extended fifth-order nonlinear equation, medium equal width equation and Caudrey–Dodd–Gibbon equation

About the solutions of Modified Lorenz system

A Modified Lorenz system is a system of three differential equations. The third differential equation in the Lorenz system is replaced with a fifth-order linear homogeneous differential equation with constant coefficients. In the paper [B. Zlatanovska and D. Dimovski, A modified Lorenz system: Definition and solution, Asian-Eur. J. Math. 13(08) (2020) 2050164], the Modified Lorenz system is defined by offering a solution, but without solving the fifth-order linear homogeneous differential equation with constant coefficients. This differential equation is solved by solving the characteristic equation of fifth degree. A formula for solving the algebraic equation of the fifth degree does not exist. Therefore, solutions for the special cases of the characteristic equation of fifth degree will be offered. The solution of the Modified Lorenz system will be completed using these forms of the characteristic equation of the fifth degree.

On Variational Symmetries and Conservation Laws of a Fifth-order Partial Differential Equation

On Variational Symmetries and Conservation Laws of a Fifth-order Partial Differential Equation

Dispersive decay bound of small data solutions to Kawahara equation in a finite time scale

In this article, we prove that small localized data yield solutions to Kawahara-type equations which have linear dispersive decay on a finite time scale depending on the size of the initial data. We use the similar method used by Ifrim and Tataru to derive the dispersive decay bound of the solutions to the KdV equation, with some steps being simpler. This result is expected to be the first result of the small data global bounds of the fifth-order dispersive equations with quadratic nonlinearity.

Fusion and fission phenomena in a (2+1)-dimensional Sawada-Kotera type system

In this study, we extend the generalized multilinear variable separation approach to a fifth-order nonlinear evolution equation. By performing asymptotic analysis on the variable separation solution, which is composed of three lower-dimensional functions, we identify a resonant regime governing dromion-dromion/solitoff interactions. In the case of two-dromion interactions, elastic, inelastic, and completely inelastic collisions are possible, while for the dromion-solitoff interaction only inelastic and completely inelastic collisions are permitted. Furthermore, we derive two types of semi-rational solutions from the quadratic function ansatz. In particular, in the scenario of a completely resonant collision between a lump and a line-soliton pair, the lump separates from one line soliton and exists briefly before merging with the other soliton, forming a localized lump in both time and space dimensions. The fusion or fission phenomena between the dromion-dromion/solitoff interaction and the lump-line soliton interaction are shown graphically.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

Probabilistic locked mode predictor in the presence of a resistive wall and finite island saturation in tokamaks

We present a framework for estimating the probability of locking to an error field in a rotating tokamak plasma. This leverages machine learning methods trained on data from a mode-locking model, including an error field, resistive magnetohydrodynamics modeling of the plasma, a resistive wall, and an external vacuum region, leading to a fifth-order ordinary differential equation (ODE) system. It is an extension of the model without a resistive wall introduced by Akçay et al. [Phys. Plasmas 28, 082106 (2021)]. Tearing mode saturation by a finite island width is also modeled. We vary three pairs of control parameters in our studies: the momentum source plus either the error field, the tearing stability index, or the island saturation term. The order parameters are the time-asymptotic values of the five ODE variables. Normalization of them reduces the system to 2D and facilitates the classification into locked (L) or unlocked (U) states, as illustrated by Akçay et al., [Phys. Plasmas 28, 082106 (2021)]. This classification splits the control space into three regions: L̂, with only L states; Û, with only U states; and a hysteresis (hysteretic) region Ĥ, with both L and U states. In regions L̂ and Û, the cubic equation of torque balance yields one real root. Region Ĥ has three roots, allowing bifurcations between the L and U states. The classification of the ODE solutions into L/U is used to estimate the locking probability, conditional on the pair of the control parameters, using a neural network. We also explore estimating the locking probability for a sparse dataset, using a transfer learning method based on a dense model dataset.

Generalized RKM methods for solving fifth-order quasi-linear fractional partial differential equation

Abstract Fractional differential equations (FDEs) are used for modeling the natural phenomena and interpretation of many life problems in the fields of applied science and engineering. The mathematical models which include different types of differential equations are used in some fields of applied sciences like biology, diffusion, electronic circuits, damping laws, fluid mechanics, and many others. The derivation of modern analytical or numerical methods for solving FDEs is a significant problem. However, in this article, we introduce a novel approach to generalize Runge Kutta Mechee (RKM) method for solving a class of fifth-order fractional partial differential equations (FPDEs) by combining numerical RKM techniques with the method of lines. We have applied the developed approach to solve some problems involving fifth-order FPDEs, and then, the numerical and analytical solutions for these problems have been compared. The comparisons in the implementations have proved the efficiency and accuracy of the developed RKM method.

Numerical Investigation of Fractional Kawahara Equation via Haar Scale Wavelet Method

The Kawahara equation is a fifth-order dispersive equation that plays a significant role in explaining the creation of non-linear water waves in the long-wavelength region. In this research, the Kawahara equation is solved numerically using the novel Haar scale-3 wavelet method in conjunction with the collocation method. The quasilinearisation approach and the Caputo derivative are used to characterise the non-linearity and fractional behaviour of the equation, respectively. To verify that the findings obtained are legitimate, residual and error estimates are generated. A thorough comparison is made between the present solutions and the numerical findings that have already been published in the literature, which demonstrates the advantages and effectiveness of the suggested technique. The Haar wavelet method reveals a dynamic system of alternative solutions for a wide variety of physical parameters.

OPTIMAL PRICING AND PRODUCTION LOT SIZE FOR TWO RATES OF PRODUCTION WITH PRICE-SENSITIVE DEMAND, PRICE BREAK-EVEN POINT, AND PROFIT MAXIMISATION IN HIGHER ORDER EQUATION

In the present study, optimal pricing and optimal lot size production policy models with price-sensitive demand of deteriorating products are considered, taking into account two distinct production rates. It is possible to begin production at one rate and then switch to a different rate after a period of time. Such a scenario is appealing, in that a big initial stock of produced goods can be avoided by starting production at a modest pace, thus reducing the initial investment and the holding cost. Further, the fifth-order equation is obtained when the equation for optimal pricing is derived. Maximising the profit is calculated based on a fifth-order equation. Both optimal pricing and production lot size are decision variables, and optimal cycle time is also one of the decision variables for determining price break-even points. As far as information is concerned, no researcher has examined optimal pricing and production lot size policies in two-rates-of-production models for their study.The objective of the present study is to examine the optimal production, optimal pricing, and optimal cycle time to reduce the total cost and to maximise the total profit. Both price break-even point and profit maximisation are considered. An appropriate mathematical model is developed. An illustrative example is provided and numerically validated using a sensitivity analysis. Microsoft Visual Basic 6.0 was used to code the model’s outcome validation.

The numerical solutions for a class of fifth-order partial differential equations via generalized RKM methods

This paper’s RKM method is explicitly constructed for solving particular fifth-order ordinary differential equations (ODEs) [1]. The proposed method has been used to solve the system of fifth-order (ODEs) numerically by converting the fifth-order partial differential equations. The algorithm of the proposed method is introduced and then used to study the numerical problems, which showed that the approximated solutions using the proposed approach agree well with the exact solutions.

Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to $ 0 $ exponentially as $ t \rightarrow \infty $. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.