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This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l ∞ and l 2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems. [ABSTRACT FROM AUTHOR]

An artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, [u(.,0),v(.,0),w(.,0)], that can evolve into a useful approximation to a hypothetical target result [u∗(.,Tmax),v∗(.,Tmax),w∗(Tmax)], at some realistic Tmax>0. Highly non smooth target data are considered, that may not correspond to actual solutions at time Tmax. Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using 512×512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on (−Δ)q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where Tmax, and/or the nonlinearity, are not sufficiently small. [ABSTRACT FROM AUTHOR]

In this article, we employ the construction of the time-marching discontinuous Petrov-Galerkin (DPG) scheme we developed for linear problems to derive high-order multistage DPG methods for nonlinear systems of ordinary differential equations. The methodology extends to abstract evolution equations in Banach spaces, including a class of nonlinear partial differential equations. We present three nested multistage methods: the hybrid Euler method and the two- and three-stage DPG methods. We employ a linearization of the problem as in exponential Rosenbrock methods, so we need to compute exponential actions of the Jacobian that change from time step to time step. The key point of our construction is that one of the stages can be postprocessed from another without an extra exponential step. Therefore, the class of methods we introduce is computationally cheaper than the classical exponential Rosenbrock methods. We provide a full convergence proof to show that the methods are second-, third-, and fourth-order accurate, respectively. We test the convergence in time of our methods on a 2D+time semilinear partial differential equation after a semidiscretization in space. [ABSTRACT FROM AUTHOR]

The nonlinear Kaczmarz method was recently proposed to solve the system of nonlinear equations. In this paper, we first discuss two greedy selection rules, i.e., the maximum residual and maximum distance rules, for the nonlinear Kaczmarz iteration. Then, based on them, two kinds of greedy randomized sampling methods are presented. Furthermore, we also devise four corresponding greedy randomized block methods, i.e., the multiple samples-based methods. The linear convergence in expectation of all the proposed methods is proved. Numerical results show that, in some applications, including brown almost linear function and generalized linear model, the greedy selection rules give faster convergence rates than the existing ones, and the block methods outperform the single sample-based ones. [ABSTRACT FROM AUTHOR]

The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the "mathematical microscope", has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics. [ABSTRACT FROM AUTHOR]

ABSTRACTAn artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, [Formula: see text], that can evolve into a useful approximation to a hypothetical target result [Formula: see text], at some realistic [Formula: see text]. Highly non smooth target data are considered, that may not correspond to actual solutions at time [Formula: see text]. Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using [Formula: see text] pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on [Formula: see text] with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where [Formula: see text], and/or the nonlinearity, are not sufficiently small.

Background: The mechanics of heat and mass transfer via nanofluid flow across many media are currently being discussed. "Nanofluids" are fluids that include highly heat-conductive nanoparticles, and they are essential for resolving engineering problems. Under the effects of activation energy, thermal radiation, and motile microorganisms, the process of heat and mass transfer through steady nanofluid flow crosses over stretched surfaces in this scenario. Methodology: For mathematical evaluation, the system of partial differential equations (PDEs) is used to describe this physical framework. By introducing suitable similarity variables with a set of boundary conditions, this mathematical system of PDEs has become a system of ordinary differential equations (ODEs). To obtain numerical results, the MATLAB built-in program "bvp4c" is used to solve the system of first-order equations. Results: In the findings and discussion section, the resulting outcomes are thoroughly examined and visually shown. The flow rate in these systems increases due to the erratic movement of microorganisms. The graphical representation shows the impacts of involving physical factors on the microorganism, thermal, concentration, and momentum profiles. Variations/ changes in these profiles can be observed by adjusting the parametric values, as depicted in the graphs. Consequently, thermal transport is boosted by 25%. Additionally, the skin friction, Nusselt, Sherwood, and microbe density numbers are determined numerically. The findings demonstrate that increasing the magnetic field parameter causes the velocity profile to decrease, increasing the radiation parameter leads to an increase in temperature description, and increasing the Lewis number causes the microorganism profile's transport rate to decrease. [ABSTRACT FROM AUTHOR]

In this paper, we consider the equation -div (a(x, u,Du)=H(x, u,Du) +a0(x) |u|θ + χ{u̸=0} f(x) in Ω, with boundary conditions u = 0 on ∂Ω, where Ω is an open bounded subset of RN, 1 < p < N, -div(a(x, u,Du)) is a Leray-Lions operator defined on W1,p0 (Ω), a0 ∈ LN/p(Ω), a0 > 0, 0 < θ ≤ 1, χ{u̸=0} is a characteristic function, f ∈ LN/p(Ω) and H(x, s, ξ) is a Carath'eodory function such that -c0 a(x, s, ξ)ξ ≤ H(x, s, ξ) sign(s) ≤ γ a(x, s, ξ)ξ a.e. x ∈ Ω, ∀s ∈ R, ∀ξ ∈ RN. For ∥a0∥N/p and ∥f∥N/p sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function exp(δ|u|) - 1 belongs to W1,p0 (Ω) for some δ ≥ γ. This solution satisfies some a priori estimates in W1,p0 (Ω). Keywords: nonlinear problems, existence, singularity. [ABSTRACT FROM AUTHOR]

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics. [ABSTRACT FROM AUTHOR]

This paper demonstrates, a numerical method to solve the one and two dimensional Burgers' equation involving time fractional Atangana-Baleanu Caputo (ABC) derivative with a non-singular kernel. The numerical stratagem consists of a quadrature rule for time fractional (ABC) derivative along with Haar wavelet (HW) approximations of one and two dimensional problems. The key feature of the scheme is to reduce fractional problems to the set of linear equations via collocation procedure. Solving the system gives the approximate solution of the given problem. To verify the effectiveness of the developed method five numerical examples are considered. Besides this, the obtained simulations are compared with some published work and identified that proposed technique is better. Moreover, computationally the convergence rate in spatiotemporal directions is presented which shows order two convergence. The stability of the proposed scheme is also described via Lax-Richtmyer criterion. From simulations it is obvious that the scheme is quite useful for the time fractional problems.

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.

This paper is inspired by an imaging problem encountered in the framework of Electrical Resistance Tomography involving two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. We consider nonlinear constitutive relationships which, at a given point in the space, present a behavior for large arguments that is described by monomials of order p and q. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted p Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the p Laplacian in inverse problems with nonlinear materials. Moreover, when p = 2, this provides a powerful bridge to bring all the imaging methods and algorithms developed for linear materials into the arena of problems with nonlinear materials. The main result of this work is that for "large" Dirichlet data in the presence of two materials of different order (i) one material can be replaced by either a perfect electric conductor or a perfect electric insulator and (ii) the other material can be replaced by a material giving rise to a weighted p Laplace problem. [ABSTRACT FROM AUTHOR]

This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted p0-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the p0-Laplacian in inverse problems with nonlinear materials. Moreover, when p0 = 2, this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for "small" Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted p0-Laplace problem. [ABSTRACT FROM AUTHOR]

For the numerical simulation of time-dependent problems, recent works suggest the use of a time marching scheme based on a tensorial decomposition of the time axis. This time-separated representation is straightforwardly introduced in the framework of the Proper Generalized Decomposition (PGD). The time coordinate is transformed into a multi-dimensional time through new separated coordinates, the micro and the macro times. From a physical viewpoint, the time evolution of all the quantities involved in the problem can be followed along two time scales, the fast one (micro-scale) and the slow one (macro-scale). In this paper, the method is applied to compute the quasi-static response of an elasto-plastic structure under cyclic loading. The study shows the existence of a physically consistent temporal decomposition in computational cyclic plasticity. Such micro-macro characterization may be particularly appealing in high-cycle loading analyses, such as aging and fatigue, addressed in a future work in progress. • Quasi-static elasto-plasticity with cyclic loading. • Proper Generalized Decomposition for time-separated representations. • Temporal evolution decomposed in a fast (micro) and a slow (macro) response. [ABSTRACT FROM AUTHOR]

In this paper, we consider the equation with boundary conditions where is an open bounded subset of is a Leray-Lions operator defined on is a characteristic function, and is a Carathéodory function such thatsign Forand sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function belongs tofor some This solution satisfies some a priori estimates in

This document is a preface to a special issue of the journal "Mathematics in Engineering, Science & Aerospace" focused on mathematical problems in science, engineering, and aerospace. The preface explains that mathematical analysis encompasses various disciplines and is applicable to fields such as engineering, economics, and biology. The special issue includes research articles on mathematical analysis with applications in engineering and applied sciences. The preface also acknowledges the contributors and reviewers who helped develop the issue. [Extracted from the article]

The main objective of this work is to present a modification of the Mittag- Leffler function to deduce a relatively new analytical approximate method (for short MMLFM) able to solve time-fractional nonlinear partial differential equations (PDEs). Moreover, we employ the MMLFM to solve the time-fractional coupled Korteweg--de Vries (KdV) model described by two nonlinear fractional partial differential equations (FPDEs) based upon Caputo fractional derivative (CFD). The simulation of projected results is presented in some figures and tables. Furthermore, we compare our solutions when α = 1 with known exact solutions which indicate a good agreement, in addition, we compare our outcomes with the results obtained by other methods in the literature such as the Natural decomposing method (NDM) and homotopy decomposition method (HDM) in order to prove the reliability and efficiency of our used method. Also, we display solutions with different values of α to present the effect of the fractional order on the proposed problem. The results of this article reveal the advantages of the MMLFM, which is simple, reliable, accurate, needs simple mathematical computations, is rapidly convergent to the exact solution, have a straightforward and easy algorithm compared to other analytical methods to study linear and nonlinear FPDEs, which makes this technique suited for real industrial or medical applications. [ABSTRACT FROM AUTHOR]

The existence of infinitely many weak solutions to a class of nonlinear elliptic Navier boundary value problems containing the p-triharmonic operator is considered by a variational approach and critical point theory. [ABSTRACT FROM AUTHOR]

In this article, we considered a 3D symmetric flow of a ternary hybrid nanofluid flow (THNF) past a nonlinear stretching surface. The effect of the thermal radiation is considered. The THNF nanofluid SiO 2 +Cu+MoS 2 /H 2 O is considered in this work, where the shapes of the particles are assumed as blade, flatlet, and cylindrical. The problem is formulated into a mathematical model. The modeled equations are then reduced into a simpler form with the help of suitable transformations. The modeled problem is then tackled with a new machine learning approach known as a hybrid cuckoo search-based artificial neural network (HCS-ANN). The results are presented in the form of figures and tables for various parameters. The impact of the volume fraction coefficients ϕ 1 , ϕ 2 , and ϕ 3 , and the radiation parameter is displayed through graphs and tables. The higher numbers of the radiation parameter (R d) and the cylinder-shaped nanoparticles, ϕ 3 , enhance the thermal profile. In each case, the residual error, error histogram, and fitness function for the optimization problem are presented. The results of the HCS-ANN are validated through mean square error and statistical graphs in the last section, where the accuracy of our implemented technique is proved. [ABSTRACT FROM AUTHOR]

In this article, we focus on a periodic resource allocation problem applied on a dynamical system which comes from a biological system. More precisely, we consider a system with N$$ N $$ resources and N$$ N $$ activities, each activity use the allocated resource to evolve up to a given time T>0$$ T>0 $$ where a control (represented by a given permutation) will be applied on the system to reallocate the resources. The goal is to find the optimal control strategies which optimize the cost or the benefit of the system. This problem can be illustrated by an industrial biological application, namely, the optimization of a mixing strategy to enhance the growth rate in a microalgal raceway system. A mixing device, such as a paddle wheel, is considered to control the rearrangement of the depth of the algae cultures, hence the light perceived at each lap. We prove that if the dynamics of the system is periodic, then the period corresponds to one reallocation whatever the order of the involved permutation matrix is. A nonlinear optimization problem for one reallocation process is then introduced. Since N!$$ N! $$ permutations need to be tested in the general case, it can be numerically solved only for a limited number of N$$ N $$. To overcome this difficulty, we introduce a second optimization problem which provides a suboptimal solution of the initial problem, but whose solution can be determined explicitly. A sufficient condition to characterize cases where the two problems have the same solution is given. Some numerical experiments are performed to assess the benefit of optimal strategies in various settings. [ABSTRACT FROM AUTHOR]

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With the increasing complexity of mechanical equipment, the control effectiveness of traditional intelligent control systems can no longer meet the needs of modern industrial production. In order to reduce errors in intelligent control systems while ensuring system performance, this study proposes a new Particle Swarm Optimization (PSO) optimization scheme. The study simplified the PSO algorithm from three aspects: algorithm parameters, speed, and position formula, and corrected the formulas for individual optimal values and global optimal values. Research will name the optimized algorithm Modified PSO (MPSO). On the basis of the MPSO algorithm, neural network intelligent control has been innovatively improved. In the experimental results, the MPSO optimized controller controlled the error within 0.01 within 0.02 seconds. At this time, the Whale Optimization Algorithm (WOA) optimized error was 0.072, and the PSO optimized error was 0.478. Compared to PSO and WOA, the control error of MPSO has decreased by 98.95% and 93.06%, respectively. In addition, the proposed method not only has the best control effect, but also has the shortest system response time, with an average time of 1.294 seconds. Compared to PSO and WOA optimization, it reduces by 61.48% and 43.07%, respectively.The results verified that the proposed method in this study can effectively improve the accuracy of intelligent control and control the error within the target range within 0.02 seconds. The research not only simplifies the calculation of the PSO algorithm, but also effectively reduces the error of the algorithm, providing a reference for research in the field of intelligent control.

We introduce an algorithm to find feedback Nash equilibria of a stochastic differential game. Our computational approach is applied to analyze optimal policies to nurture a romantic relationship in the long term. This is a fundamental problem for the applied sciences, which is naturally formulated in this work as a stochastic differential game with nonlinearities. We use our computational model to analyze the risk of marital breakdown. In particular, we introduce the concept of “love at risk,” which allows us to estimate the probability of a couple breaking up in the face of possible unfavorable scenarios. [ABSTRACT FROM AUTHOR]

In this work, we propose a two-point Landweber-type method with general convex penalty terms for solving nonsmooth nonlinear inverse problems. The design of our method can cope with nonsmooth nonlinear inverse problems or the nonlinear inverse problems whose data are contaminated by various types of noise. The method consists of the two-point acceleration strategy and inner solvers. Inner solvers are used to solve the minimization problems with respect to the penalty terms in each steps. If the minimization problems can be solved explicitly, the inner solvers will be chosen to be the exact solvers. Otherwise, we will use inexact solvers as inner solvers. Convergence results are given without utilizing the Gâteaux differentiability of the forward operator or the reflexivity of the image space. Numerical simulations are given to test the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a mathematical model for the evaluation of temperature of a surface covered with a thin heat-insulating layer according to the data of measurements of temperature on the free surface of the coating and in the ambient medium. The model involves the mechanisms of conductive and radiation energy transfer in the volume of the layer, conductive and radiation heat exchange with the surface covered by the layer, convective and radiation heat exchange with the ambient medium on the free surface of the coating capable of emission, absorption, and reflection of thermal electromagnetic radiation. We also present the results of numerical analysis of the solutions of nonlinear problem based on the developed iterative procedure. [ABSTRACT FROM AUTHOR]

In this paper, the stability of a class of Liu–Wang's optimal eighth-order single-parameter iterative methods for solving simple roots of nonlinear equations was studied by applying them to arbitrary quadratic polynomials. Under the Riemann sphere and scaling theorem, the complex dynamic behavior of the iterative method was analyzed by fractals. We discuss the stability of all fixed points and the parameter spaces starting from the critical points with the Mathematica software. The dynamical planes of the elements with good and bad dynamical behavior are given, and the optimal parameter element with stable behavior was obtained. Finally, a numerical experiment and practical application were carried out to prove the conclusion. [ABSTRACT FROM AUTHOR]

This study explored the convergence rate of the hybrid numerical iterative technique (HNIT) for the solution of nonlinear problems (NLPs) of one variable (f (x) = 0). It is sightseen that convergence rate is two for the HNIT. By the HNIT, several algebraic and transcendental NLPs of one variable have been illustrated as an approximate real root for efficient performance. In many instances, HNIT is more vigorous and attractive than well-known conventional iterative techniques (CITs). The computational tool MATLAB has been used for the solution of iterative techniques. [ABSTRACT FROM AUTHOR]

In this article, we considered a 3D symmetric flow of a ternary hybrid nanofluid flow (THNF) past a nonlinear stretching surface. The effect of the thermal radiation is considered. The THNF nanofluid SiO2+Cu+MoS2/H2O is considered in this work, where the shapes of the particles are assumed as blade, flatlet, and cylindrical. The problem is formulated into a mathematical model. The modeled equations are then reduced into a simpler form with the help of suitable transformations. The modeled problem is then tackled with a new machine learning approach known as a hybrid cuckoo search-based artificial neural network (HCS-ANN). The results are presented in the form of figures and tables for various parameters. The impact of the volume fraction coefficients ϕ1, ϕ2, and ϕ3, and the radiation parameter is displayed through graphs and tables. The higher numbers of the radiation parameter (Rd) and the cylinder-shaped nanoparticles, ϕ3, enhance the thermal profile. In each case, the residual error, error histogram, and fitness function for the optimization problem are presented. The results of the HCS-ANN are validated through mean square error and statistical graphs in the last section, where the accuracy of our implemented technique is proved.

Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized- α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems. [ABSTRACT FROM AUTHOR]

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering. [ABSTRACT FROM AUTHOR]

Adequate mathematical models and computational algorithms are developed in this study to investigate specific features of the deformation processes of elastic rotational shells at large displacements and arbitrary rotation angles of the normal line. A finite difference method (FDM) is used to discretize the original continuum problem in spatial variables, replacing the differential operators with a second-order finite difference approximation. The computational algorithm for solving the nonlinear boundary value problem is based on a quasi-dynamic form of the ascertainment method with the construction of an explicit two-layer time-difference scheme of second-order accuracy. The influence of physical and mechanical characteristics of isotropic and composite materials on the deformation features of elastic spherical shells under the action of surface loading of "tracking" type is investigated. The results of the studies conducted have shown that the physical and mechanical characteristics of isotropic and composite materials significantly affect the nature of the deformation of the clamped spherical shell in both the subcritical and post-critical domains. The developed mathematical models and computational algorithms can be applied in the future to study shells of rotation made of hyperelastic (non-linearly elastic) materials and soft shells. [ABSTRACT FROM AUTHOR]

Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction of the proposed method is derived based on a modified conjugate gradient method, in such a way that it is sufficiently descent. It is worth noting that, unlike many existing methods that require a monotonicity assumption to prove the convergence result, our new method needs the underlying function to be pseudomonotone, which is a weaker assumption. The performance of the proposed algorithm is demonstrated on a set of some test problems and applications arising from compressive sensing. The obtained results confirm that the proposed method is effective compared to some existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

An inverse problem of finding a temperature-dependent thermal conductivity of a substance is considered. The analysis is based on the first boundary value problem for the three-dimensional nonstationary heat equation. The sample of the substance under investigation has the form of a rectangular parallelepiped. The inverse coefficient problem is reduced to a variational problem. The root-mean-square deviation of the calculated heat flux on the surface of the body from the experimentally obtained flux is chosen as the cost functional. The paper investigates the possibility of solving the variational problem by optimization methods of the second order of convergence. On the example of a number of nonlinear problems whose coefficients are temperature-dependent, a comparative analysis of the solution of these problems by means of the gradient method and the Levenberg–Marquardt method is performed. The accuracy of calculating the elements of the Jacobi-type matrix required to implement the Levenberg–Marquardt method has a significant impact on the convergence of the iterative process. It is essential that in our approach the elements of the Jacobi-type matrix are calculated with machine precision due to the use of the fast automatic differentiation technique. Much attention is paid to the features of solving the inverse problem associated with its three-dimensional spatial nature. [ABSTRACT FROM AUTHOR]

Optimal control is seen by researchers from a different perspective than that from which the industry practitioners see it. Either type of user can easily become confounded when deciding which manner of optimal control should be used for guidance and control of mechanics. Such optimization methods are useful for autonomous navigation, guidance, and control, but their performance is hampered by noisy multi-sensor technologies and poorly modeled system equations, and real-time on-board utilization is generally computationally burdensome. Some methods proposed here use noisy sensor data to learn the optimal guidance and control solutions in real-time (online), where non-iterative instantiations are preferred to reduce computational burdens. This study aimed to highlight the efficacy and limitations of several common methods for optimizing guidance and control while proposing a few more, where all methods are applied to the full, nonlinear, coupled equations of motion including cross-products of motion from the transport theorem. While the reviewed literature introduces quantitative studies that include parametric uncertainty in nonlinear terms, this article proposes accommodating such uncertainty with time-varying solutions to Hamiltonian systems of equations solved in real-time. Five disparate types of optimum guidance and control algorithms are presented and compared to a classical benchmark. Comparative analysis is based on tracking errors (both states and rates), fuel usage, and computational burden. Real-time optimization with singular switching plus nonlinear transport theorem decoupling is newly introduced and proves superior by matching open-loop solutions to the constrained optimization problem (in terms of state and rate errors and fuel usage), while robustness is validated in the utilization of mixed, noisy state and rate sensors and uniformly varying mass and mass moments of inertia. Compared to benchmark, state-of-the-art methods state tracking errors are reduced one-hundred ten percent. Rate tracking errors are reduced one-hundred thirteen percent. Control utilization (fuel) is reduced eighty-four percent, while computational burden is reduced ten percent, simultaneously, where the proposed methods have no control gains and no linearization.

An approach to solving the problem of determining the thermal conductivity of a substance based on the results of observing the dynamics of the temperature field is proposed. The consideration is based on the Dirichlet boundary value problem for the three-dimensional nonstationary heat equation. The effectiveness of the proposed approach is based on the application of the Fast Automatic Differentiation technique. The required thermal conductivity coefficient is determined as the solution to the formulated optimal control problem. [ABSTRACT FROM AUTHOR]

Solution of large‐scale nonlinear stochastic mechanics problems such as plasticity is generally very expensive. In this work, a domain decomposition based scalable method is proposed for solving such problems. The mechanics problem and random fields are discretized using finite element (FE) bases and Karhunen–Loève expansion, respectively. The FE mesh is partitioned that offers (i) both spatial and stochastic dimensionality reduction and (ii) an inherent framework for parallelization. Then a stochastic collocation based surrogate model is built for each subdomain wherein at each collocation point a deterministic nonlinear problem is solved. The deterministic nonlinear problem is solved using Newton–Raphson method. The linear system of equations involving the Jacobian is solved using the dual‐primal version of the FE tearing and interconnecting (FETI‐DP) method, due to its demonstrated scalability for deterministic problems. Stochastic collocation and FETI‐DP are inherently and independently parallelizable. Finally, at the post‐processing stage, a statistical sampling from the surrogate model is performed by preserving the structure of the input random field. The proposed method is numerically tested for p‐Laplace and plain‐strain plasticity problems, and found to be computationally efficient and accurate. In parallel implementation, the method showed good scalability. [ABSTRACT FROM AUTHOR]

This study aims to investigate the deployment of a proposed search field division method (SFDM) within evolutionary algorithms (EAs) to enhance the capability of searching for the global optima in nonlinear problems. The proposed technique is benchmarked against the following eight widely-used single-modal, multi-modal, and unimodal benchmark functions: Sphere, Rosenbrock, Rastringin, Griewank, Ackley, Fletcher, Quartic, and Schwefel functions, and the outcome is compared to their standard EAs counterparts to validate the effectiveness of the deployed approach in EAs. In the proposed method, we apply three low, medium, high field divisions (1, 2, and 5) dimensions on nine different EAs simultaneously with two different scenarios, 10 and 100 variables, to reach the optimal solution. Then for the validity of our proposed SFDM technique, we examined the exploration-exploitation search space rates and diversity behavior. The results of the implementation of SFDM on eight benchmark test functions show that the consideration of dimensions using SFDM for EAs improves the outcomes of all nine tested EAs. In our proposed method, we find better compatibility with the integration of SFDM in the Particle Swarm Optimization Algorithm concerning searching for the optimum solution relative to the other EAs. A novel Search Field Division Method (SFDM) for evolutionary algorithms. Applying three field divisions (1, 2, and 5) dimensions on nine different Evolutionary Algorithms (EAs) simultaneously to reach the optimal global solution. The implementation of SFDM on eight benchmark functions shows significant improvement in all nine tested EAs. In our proposed method, we find the Particle Swarm Optimization Algorithm has better compatibility with respect to the other EAs. [ABSTRACT FROM AUTHOR]

To solve nonlinear problems, we construct two kinds of greedy capped nonlinear Kaczmarz methods by setting a capped threshold and introducing an effective probability criterion for selecting a row of the Jacobian matrix. The capped threshold and probability criterion are mainly determined by the maximum residual and maximum distance rules. The block versions of the new methods are also presented. We provide the convergence analyses of these methods and their numerical results behave quite well. [ABSTRACT FROM AUTHOR]