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This article studies the output regulation problem (ORP) for nonlinear systems based on prescribed performance control (PPC). The items with the partial derivative of the virtual controller are combined together by using backstepping, and then the fuzzy logic systems (FLSs) are used to approximate these combined items, so that the designed virtual controller does not have the partial derivative of the previous virtual controllers. Therefore, this method not only reduces the calculation burden in the backstepping method, but also avoids the disadvantages of dynamic surface control (DSC). Finally, a function ϴ is constructed such that the overall performance (dynamic performance and steady-state performance) of the tracking error (OPTE) is constrained by PP functions. The proposed control algorithm ensures that all the signals are semi-globally uniformly ultimately bounded (SGUUB), and the tracking error achieves the PPC. Simulation examples are provided to illustrate the effectiveness of the proposed method.

The finite-difference scheme of directed differences (the Courant-Isaacson-Ries scheme), which is widely used in the practice of aerohydrodynamic calculations, is studied theoretically and on the example of test problems. We applied the commonly used in practice Courant-Isakson-Ries directional difference scheme that allowed us to find and show distributions of velocities where the laws of the matter conservation are violated in the calculations in solving the matter conservation equations or the correspondence of the obtained solutions to the most general practical understandings on the essence of the matter transfer. A scheme free from the shortcomings of the Courant-Isaacson-Ries scheme has been constructed, tested, and proposed for use in aerohydrodynamic calculations by the finite difference method. Moreover, all the valuable properties of this well-known scheme are preserved. Among the maintained properties: transportability, conservatism, stability in calculations, invariance, adequacy of the essence of the physical phenomenon of the transfer of matter in space. The disadvantages of the new finite-difference scheme proposed for solving the equations of conservation of matter should be considered: an increase in the required RAM for storing electronic means of calculating information about the velocity field in memory and an increase in the number of calculations needed. However, this is an insignificant price to pay for guaranteed conservatism in calculations while maintaining all other valuable properties of calculation schemes. A more stable behavior of the solution in the calculations will allow increasing the time steps so much that the losses in the number of calculations become not noticeable., {"references":["1.\tThiele, R. (2005). The Mathematics and Science of Leonhard Euler. In: VanBrummelen, G., Kinyon, M. 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Mathematics. 9, 2920. https://doi.org/10.3390/ math9222920","11.\tEmmanuel, Jacob & Victor, Alexander. (2020). Numerical Solution for Weather Forecasting Using Finite Difference Scheme. IOSR Journal of Mathematics. 16. 49-56. DOI:10.9790/5728-1603054956.","12.\tKantha, L, Clayson, C A (2000) Numerical Models of Oceans and Oceanic Processes, Int. GeophysAcademic PressISBN: 0-12-434068-7","13.\tAlosaimi, M, Lesnic, D, Johansson, B T. (2021) Solution of the Cauchy problem for the wave equation using iterative regularization, Inverse Problems in Science and Engineering, 29:13, 2757-2771, DOI: 10.1080/17415977.2021.1949590.","14.\tMickens, R.E. (2020). Nonstandard Finite Difference Schemes. Methodology and Applications. https://doi.org/10.1142/11891","15.\tMoiseev, N Y, Silanteva, I Y. (2008) Difference schemes of arbitrary order of approximation for solving linear transport equations with constant coefficients by the Godunov method with antidiffusion. Journal of Computational Mathematics and Mathematical Physics, Volume 48, No7, pp 1282-1293","16.\tBakirov, K B, Duishokov, K D. (2003) Numerical methods of weather forecasting. Quasi-geostrophic barotropic model of the atmosphere. Textbook, Meteorology, Bishkek, 39.","17.\tTkachev, D, Blokhin, A. Courant-Friedrichs' Hypothesis and Stability of the Weak Shock Wave Satisfying the Lopatinski Condition Open Journal of Applied Sciences Vol.3 No.1B1, July 12, 2013 DOI: 10.4236/ojapps.2013.31B1016.","18.\tMarchuk, G.I., Kurbatkin, G.P. (1978). Numerical weather forecast Earth and Universe. Science, Novosibirsk, pp 37-43.","19.\tTsydenov, B O, Starchenko, A V. (2011). Numerical simulation of the thermal bar effect in Lake Baikal during the spring-summer warming period. Vestn.Tomsk.State Univ.Mathematics and Mechanics 1(13): 120-129.","20.\tChang, S Y (2022) A novel series of solution methods for solving nonlinear stiff dynamic problems, Nonlinear Dynamics, DOI:10.1007/s11071-021-07048-0.","21.\tPrusov, V A, Doroshenko, A E, Chernysh, R I et al. (2007) Efficient difference scheme for numerical solution of a convective diffusion problem. CybernSyst Anal 43, 368–376 https://doi.org/10.1007/s10559-007-0058-2","21.\tPrusov, V A, Doroshenko, A E, Chernysh, R I et al. (2007) Efficient difference scheme for numerical solution of a convective diffusion problem. CybernSyst Anal 43, 368–376 https://doi.org/10.1007/s10559-007-0058-2","23.\tPopov, I V, Fryazinov, I V. (2015) The method of adaptive artificial viscosity for the numerical solution of equations of gas dynamics. Krasand, Moscow, 275.","24.\tZalesny, V B, Gusev, A. (2009). Mathematical model of the World Ocean dynamics with algorithms of variational assimilation of temperature and salinity fields. 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During a severe accident, steam generator (SG) tubes undergo rapid changes in the pressure and temperature. Therefore, an appropriate creep model to predict a short term creep damage is essential. In this paper, a novel creep model for Alloy 690 SG tube material was proposed. It is based on the theta (θ) projection method that can represent all three stages of the creep process. The original θ projection method poses a limitation owing to its inability to represent experimental creep curves for SG tube materials for a large strain rate in the tertiary creep region. Therefore, a new modified θ projection method is proposed; subsequently, a master curve for Alloy 690 SG material is also proposed to optimize the creep model parameters, θi (i = 1–5). To adapt the implicit creep scheme to the finite element code, a partial derivative of incremental creep with respect to the stress is necessary. Accordingly, creep model parameters with a strictly linear relationship with the stress and temperature were proposed. The effectiveness of the model was validated using a commercial finite element analysis software. The creep model can be applied to evaluate the creep rupture behavior of SG tubes in nuclear power plants.

Distributed stabilization problem is studied in this article for multiple heterogeneous agents in the uncertain strong-weak competition network with exogenous disturbances, where the agents are modeled by the second-order systems with different nonlinear intrinsic dynamics, and the network uncertainty is characterized by unknown nonzero parameters, which contains three different relationships among agents: 1) cooperation; 2) strong competition; and 3) weak competition. To achieve distributed stabilization, the whole network is first divided into two parts: 1) identifiable part and 2) unidentifiable part, and a new distributed robust integral sign of the error (RISE) controller is designed for each agent, where the selection rules of the corresponding parameters are given. It is shown that the heterogeneous multiagent system (MAS) can achieve distributed stabilization no matter whether the identifiable part is structurally balanced or not. Furthermore, it is proved that the global distributed stabilization is achieved for the heterogeneous agents provided that the partial derivatives of the nonlinear intrinsic dynamics are bounded. Finally, two numerical examples are given to demonstrate the effectiveness of the designed controller.

In this paper, a fixed-time controller under the mechanism of event-trigger is designed for a class of nonlinear pure-feedback systems with non-differentiable non-affine functions. By properly modeling non-affine terms, the limitation of the partial derivatives of non-affine functions is eliminated. In our design process, we first develop a fixed-time adaptive controller using decoupling method. Then, a relative threshold event-trigger mechanism (ETM) is introduced in Section 3.1. The proposed controller can not only stabilize the system within a fixed-time, but also save communication resources more effectively. Lastly, the feasibility of the proposed control scheme is verified by two simulation examples.

Empirical distributions in a range of fields are often substantially non-Gaussian but smooth enough to suggest that the underlying population distribution has at least several derivatives. Linear combinations of many random variables often have smooth densities even if the random variables are not independent. The main result here generalizes the elementary result that a convolution of densities has as many derivatives as the sum of the number of derivatives of each component to certain nonlinear functions of random vectors whose components may not be independent. Linearity is replaced by an assumption that the nonlinear function has positive first partial derivatives bounded away from 0 along at least some coordinates and independence is replaced by an assumption that the joint density has certain mixed partial derivatives. This approach to justifying the smoothness of empirical distributions is contrasted with other possible approaches, including solutions of stochastic partial differential equations and ergodic distributions for chaotic dynamical systems.

In the neural network adaptation, the Newton method could find a minimum with its second-order partial derivatives, and convergent gradient steepest descent could assure its error convergence with its time-varying adaptation rates. In this article, the convergent Newton method is proposed as the combination of the Newton method and the convergent gradient steepest descent for the neural networks adaptation, where the convergent Newton method incorporates the second-order partial derivatives inside of the time-varying adaptation rates. Hence, the convergent Newton method could assure its error convergence and could find a minimum. Experiments show that the convergent Newton method obtains satisfactory results in the electric energy usage data prediction.

An analytical approach about the general type of step-index optical fiber modes and their variations is presented in this paper. The theoretical model is solved through our analytical method. With the boundary conditions, a composite equation set is established and solved. The precise results of all modes are achieved with a high-precision numerical solution. The modal characteristic contains the distributions of electromagnetic fields, the effective refractive index, and its tendencies. The modal sensitivity is based on the former and the partial derivative of the effective refractive index (neff/ni and neff/ri). The derivative rule of implicit functions and determinant function allows our method to analyze the sensitivity of modes. Furthermore, four qualitative deductions are derived during the process, including the qualitative analysis of modes and their constraints, supporting previous experiments. Our work is beneficial for exploiting new fiber and researching relative fiber sensors.

The non-linear conjugate gradient method is a very effective technique for addressing Large-Scale minimization problems, and it has a wide range of applications in Mathematics, Chemistry, Physics, Engineering, and Medicine, etc. In this paper, we present a new spectral conjugate gradient algorithm, a non-linear conjugate gradient algorithm, whose derivation is based on the Hisham–Khalil (KH) and Newton algorithms Based on Pure Conjugacy Condition, The importance of the research lies in finding an appropriate way to solve all kinds of linear and non-linear fuzzy equations because the Buckley and Qu's method is ineffective in solving all kinds of fuzzy equations and because the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in the solution. The descent property of the suggested method is shown provided that the step size meets the strong Wolfe conditions. In many circumstances, numerical results demonstrate that the novel technique is more efficient than the Fletcher–Reeves (FR) and Hisham–Khalil (KH) procedures in solving Fuzzy Nonlinear Equations.

Inversion of large-scale time-domain transient electromagnetic (TEM) surveys is computationally expensive and time-consuming. The calculation of partial derivatives for the Jacobian matrix is by far the most computationally intensive task, as this requires calculation of a significant number of forward responses. We propose to accelerate the inversion process by predicting partial derivatives using an artificial neural network. Network training data for resistivity models for a broad range of geological settings are generated by computing partial derivatives as symmetric differences between two forward responses. Given that certain applications have larger tolerances for modeling inaccuracy and varying degrees of flexibility throughout the different phases of interpretation, we present four inversion schemes that provide a tunable balance between computational time and inversion accuracy when modeling TEM datasets. We improve speed and maintain accuracy with a hybrid framework, where the neural network derivatives are used initially and switched to full numerical derivatives in the final iterations. We also present a full neural network solution where neural network forward and derivatives are used throughout the inversion. In a least-squares inversion framework, a speedup factor exceeding 70 is obtained on the calculation of derivatives, and the inversion process is expedited ~36 times when the full neural network solution is used. Field examples show that the full nonlinear inversion and the hybrid approach gives identical results, whereas the full neural network inversion results in higher deviation but provides a reasonable indication about the overall subsurface geology.

We present a method based on integrated radial basis function-finite difference for numerical solution of plane elastostatic equations which is a boundary value problem. The two-dimensional version of the governed equation is solved by the proposed method on various geometries such as the rectangular and irregular domains. In the current paper, one of our goals is to present an improved integrated radial basis function method based on the finite difference technique to approximate the second-order mixed partial derivatives with respect to x and y to get more accurate numerical results. Several examples are solved by applying integrated radial basis function based on finite difference method to check its accuracy and validity.

This article mainly solves the consensus issue of parabolic partial differential equation (PDE) agents with switching topology by output feedback. A novel edge-based adaptive control protocol is designed to reach consensus under the condition that the switching graphs are always connected at any switching instants. Different from the existing adaptive protocol associated with partial differential dynamics, the proposed adaptive observer-type law relies on the relative output information rather than relative state information. A proper Lyapunov functional is constructed and some important lemmas are used, then a sufficient condition is obtained for the consensus of parabolic PDE agents on switching graphs. Besides, a corollary about the distributed adaptive consensus of parabolic PDE agents on fixed undirected communication networks is given. Finally, the theoretical results are demonstrated by two numerical simulations.

This article is concerned with the existence of entire solutions for the following complex second order partial differential-difference equation \begin{document}$ \left(\frac{\partial^2 f(z_1, z_2)}{\partial z_1^2}+\frac{\partial^2 f(z_1, z_2)}{\partial z_2^2}\right)^{l}+f(z_1+c_1, z_2+c_2)^{k} = 1, $\end{document} where $ c_1, c_2 $ are constants in $ \mathbb{C} $ and $ k, l $ are positive integers. In addition, we also investigate the forms of finite order transcendental entire solutions for several complex second order partial differential-difference equations of Fermat type, and obtain some theorems about the existence and the forms of solutions for the above equations. Meantime, we give some examples to explain the existence of solutions for some theorems in some cases. Our results are some generalizations of the previous theorems given by Qi [23], Xu and Cao [35], Liu, Cao and Cao [17].

Recent advances in the time-domain electromagnetic (TDEM) method have dramatically improved detection and discrimination of subsurface targets. Inversion of observed response using a 3-D orthogonal magnetic dipolar model provides location, orientation, and intrinsic responses of the target based on deterministic optimization methods, which is dependent on the initial values and could be trapped in local minimum solutions. In this letter, we applied a supervised descent method (SDM) to the inversion of electromagnetic induction (EMI) data accurately by individually and simultaneously training every single sample in the training set to avoid the direct use of the SDM that causes inaccurate classification results. This method provides a new way to incorporate prior information using gradient learning and reduce the computational complexity as it does not compute partial derivatives in the nonlinear least-squares problem or groups difference in the heuristic random search algorithm. Then the simulation and field experiments are performed to verify the feasibility of this method. Both the simulation and experimental results demonstrate that the SDM shows good performance and robustness in the classification of subsurface anomalous targets.

This paper investigated the adaptive fuzzy finite-time control problem for a class of high-order stochastic nonlinear systems with a class of exponential type prescribed performance function. It is assumed that the nonlinear functions in the controlled plant are unknown, in which fuzzy logic systems (FLSs) are utilized due to the approximation ability of any unknown continuous functions with arbitrary approximation errors. Based on the FLSs and backstepping design technique, an novel adaptive fuzzy tracking control strategy is proposed to guarantee that the closed-loop nonlinear system is semi-globally finite-time stable in probability via Lyapunov stability theory and It ${\hat o}$ formula. Compared with existing results, the transformed error signal was regarded as a stochastic variable. In addition, the expressions of the first and second-order partial derivatives of the transformed error signals are given in this paper. Finally, a simulation example with different covariance values is given to show the effectiveness of the proposed control strategy.

In this work, a novel Robin transmission condition (RTC) enhanced discontinuous Galerkin (DG) method is proposed for the DC IR-Drop analysis of power distribution networks with Joule Heating effects included. Unlike the conventional DG method, the proposed DG method straightforwardly applied to discretize the second-order spatial partial differential governing equations for the electrostatic potential Φ and the steady-state temperature T, respectively. The numerical flux in DG used to facilitate the information exchange among neighboring subdomains introduces two additional variables: the current density J for the electrical potential equation and the thermal flux q for the thermal equation. To solve them, at the interface of neighboring subdomains a RTC is presented as the second equation to establish another connection for solutions in neighboring subdomains. With this strategy, the number of degrees of freedom (DoF) involved in the proposed RTC-DG method is dramatically reduced compared with the traditional DG method. The finalized matrix system is solved in a FETI-like procedure, namely, the unknowns are obtained in a subdomain-by-subdomain scheme. Finally, the accuracy and the efficiency of the proposed RTC-DG method is validated by serval representative examples.

In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

A Robin boundary sub-diffusion equation is considered with fractional partial derivatives of the Caputo type. The model is an extension of various well-known equations from mathematical physics, biology, and chemistry. Initial–boundary data are imposed upon a closed and bounded spatial domain. We state and prove two main theorems in differential and difference settings to ensure the algebraic decay rate of the long-time behavior for that kind of problem. The dissipation of the continuous solution for such a problem is discussed in the first theorem based on energy inequalities and by the aid of Gronwall inequalities. It demonstrates that with an L 2 ( Ω ) -bounded absorbing set, the solution is dissipated with respect to time. The numerical dissipativity is proved in the second theorem by using discrete energy inequalities and the discrete Paley–Wiener inequality. Finally, an example is provided to illustrate the main outcomes.

A new event-triggered iterative learning control method is proposed for handling the distributed containment control problem of model-free multiagent systems under a fixed directed graph. The designed controller merely uses the input and output signals, controlled model information is not required. At first, the unknown dynamic is transformed into the linearization model upon the base of pseudo partial derivative. Secondly, the novel distributed containment controller is proposed for each follower by use of iterative learning algorithm. Moreover, a new trigger mechanism is designed to save energy of the systems, such that the updating number of the proposed controller can be reduced greatly. Mathematical deduction shows that the controller can render the outputs of the followers converge to a convex hull formed by the outputs of leaders. Finally, simulation examples are given for verifying the significance of proposed method.

Solving the wave equation is one of the most (if not the most) fundamental problems we face as we try to illuminate the Earth using recorded seismic data. The Helmholtz equation provides wavefield solutions that are dimensionally reduced, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion. However, our ability to attain such wavefield solutions depends often on the size of the model and the complexity of the wave equation. Thus, we use here a recently introduced framework based on neural networks to predict functional solutions through setting the underlying physical equation as a loss function to optimize the neural network (NN) parameters. For an input given by a location in the model space, the network learns to predict the wavefield value at that location, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of the Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions of the background wavefield. Providing the NN with a reasonable number of random points from the model space will ultimately train a fully connected deep NN to predict the scattered wavefield function. The size of the network depends mainly on the complexity of the desired wavefield, with such complexity increasing with increasing frequency and increasing model complexity. However, smaller networks can provide smoother wavefields that might be useful for inversion applications. Preliminary tests on a two-box-shaped scatterer model with a source in the middle, as well as, the Marmousi model with a source at the surface demonstrate the potential of the NN for this application. Additional tests on a 3D model demonstrate the potential versatility of the approach.

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.