Solution Of Nonlinear Problem Research Articles

Approximation of solutions of nonlinear problems in hyperbolic spaces

Approximation of solutions of nonlinear problems in hyperbolic spaces

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

Stabilization‐free virtual element method for 2D elastoplastic problems

AbstractIn this paper, a novel first‐ and second‐order stabilization‐free virtual element method is proposed for two‐dimensional elastoplastic problems. In contrast to traditional virtual element methods, the improved method does not require any stabilization, making the solution of nonlinear problems more reliable. The main idea is to modify the virtual element space to allow the computation of the higher‐order projection operator, ensuring that the strain and stress represent the element energy accurately. Considering the flexibility of the stabilization‐free virtual element method, the elastoplastic mechanical problems can be solved by radial return methods known from the traditional finite element framework. plasticity with hardening is considered for modeling the nonlinear response. Several numerical examples are provided to illustrate the capability and accuracy of the stabilization‐free virtual element method.

On the behaviors of solutions of a nonlinear diffusion system with a source and nonlinear boundary conditions

We study the global solvability and unsolvability of a nonlinear diffusion system with nonlinear boundary conditions in the case of slow diffusion. We obtain the critical exponent of the Fujita type and the critical global existence exponent, which plays a significant part in analyzing the qualitative characteristics of nonlinear models of reaction-diffusion, heat transfer, filtration, and other physical, chemical, and biological processes. In the global solvability case, the key components of the asymptotic solutions are obtained. Iterative methods, which quickly converge to the exact solution while maintaining the qualitative characteristics of the nonlinear processes under study, are known to require the presence of an appropriate initial approximation. This presents a significant challenge for the numerical solution of nonlinear problems. A successful selection of initial approximations allows for the resolution of this challenge, which depends on the value of the numerical parameters of the equation, which are primarily in the computations recommended using an asymptotic formula. Using the asymptotics of self-similar solutions as the initial approximation for the iterative process, numerical calculations and analysis of the results are carried out. The outcomes of numerical experiments demonstrate that the results are in excellent accord with the physics of the process under consideration of the nonlinear diffusion system.

A multiparameter mathematical model for the problem of nonlinear filtration of fluids in two-layer media

In the realm of fluid dynamics and porous media, the nonlinear filtration of fluids in two-layer media presents a complex and intriguing challenge that demands a comprehensive mathematical framework for accurate representation and analysis. This paper introduces a multiparameter mathematical model designed to elucidate the intricacies of fluid filtration in heterogeneous environments, specifically focusing on two-layer media. The authors research the construction of a multiparameter mathematical model and numerical solution of nonlinear problems of the theory of filtration of unstructured and structured fluids in hydrodynamically connected multilayer formations with different characteristics. The developed models and solution algorithms were tested on hypothetical data.

Exact Solutions of the Modified Nonlinear Burgers' Equation

The expansion approach has been used to solve the modified nonlinear Burgers' equation, which has a nonlinear convection term, a viscosity term, and a time-dependent term in its structure. In this paper, the main focus is to find exact solutions of the modified nonlinear Burgers' equation. The (G′=G)-expansion method is one of the methods used to find exact solutions of nonlinear problems. It requires an appropriate transform equation to convert partial differential equations to ordinary differential equations, making it easier to find the solution. In this work, we choose the traveling wave equation to covert the equation. The results show that the exact solutions of the modified nonlinear Burgers' equation by the (G′=G)-expansion method are decreasing if traveling wave parameter, ω, increases for the first case and the exact solution is increasing if ω is increasing for the second case. It is observed that (G′=G)-expansion method is an advanced and easy tool for finding exact solutions of the modified nonlinear Burgers' equation.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

ANALYSIS OF ONE-DIMENSIONAL NON-NEWTONIAN MODELS FOR SIMULATION OF BLOOD FLOW IN ARTERIES

The paper is devoted to the theoretical analysis of one-dimensional (1D) models of blood flow. The non-Newtonian nature of blood is taken into account. The generalized Newtonian models, in which the dynamic viscosity is dependent only on the shear rate, are considered. The models are constructed by averaging the simplified Navier–Stokes system in cylindrical coordinates. The 1D models, corresponding to Newtonian, Power Law, Carreau, Carreau–Yasuda, Cross, Simplified Cross, Yeleswarapu, Modified Yeleswarapu, and Quemada models, are compared in this paper. The comparison is performed in order to estimate the influence of non-Newtonian frictional terms, velocity profiles and hematocrit values on the deviations of the solutions from the solution for the Newtonian case. For this purpose, two problems that can be solved analytically are considered: problems for the infinite interval and finite interval with periodic conditions. For the quantitative comparison of solutions, the non-Newtonian factor, which characterizes the relative deviations of solutions from the solution for the Newtonian model, is introduced. The analytical solutions of nonlinear problems are obtained by the perturbation method. As it is demonstrated for both solved problems, the relative deviations increase with the flattening of the velocity profile. For the models that depended on hematocrit, the non-Newtonian factor values increased with the increase in hematocrit value. The solutions obtained in the paper can be used as a tool for the comparison of different 1D models of blood flow and for the testing of programs that implement numerical algorithms.

Numerical Solution of Nonlinear Problems with Multiple Roots Using Derivative-Free Algorithms

In the study of systems’ dynamics the presence of symmetry dramatically reduces the complexity, while in chemistry, symmetry plays a central role in the analysis of the structure, bonding, and spectroscopy of molecules. In a more general context, the principle of equivalence, a principle of local symmetry, dictated the dynamics of gravity, of space-time itself. In certain instances, especially in the presence of symmetry, we end up having to deal with an equation with multiple roots. A variety of optimal methods have been proposed in the literature for multiple roots with known multiplicity, all of which need derivative evaluations in the formulations. However, in the literature, optimal methods without derivatives are few. Motivated by this feature, here we present a novel optimal family of fourth-order methods for multiple roots with known multiplicity, which do not use any derivative. The scheme of the new iterative family consists of two steps, namely Traub-Steffensen and Traub-Steffensen-like iterations with weight factor. According to the Kung-Traub hypothesis, the new algorithms satisfy the optimality criterion. Taylor’s series expansion is used to examine order of convergence. We also demonstrate the application of new algorithms to real-life problems, i.e., Van der Waals problem, Manning problem, Planck law radiation problem, and Kepler’s problem. Furthermore, the performance comparisons have shown that the given derivative-free algorithms are competitive with existing optimal fourth-order algorithms that require derivative information.

A Halpern-type algorithm for a common solution of nonlinear problems in Banach spaces

Abstract In this article, we propose a Halpern-type subgradient extragradient algorithm for solving a common element of the set of solutions of variational inequality problems for continuous monotone mappings and the set of f-fixed points of continuous f-pseudocontractive mappings in reflexive real Banach spaces. In addition, we prove a strong convergence theorem for the sequence generated by the algorithm. As a consequence, we obtain a scheme that converges strongly to a common f-fixed point of continuous f-pseudocontractive mappings and a scheme that converges strongly to a common zero of continuous monotone mappings in Banach spaces. Furthermore, we provide a numerical example to illustrate the implementability of our algorithm.

DeepFEM: A Novel Element-Based Deep Learning Approach for Solving Nonlinear Partial Differential Equations in Computational Solid Mechanics

In this paper, an element-based deep learning approach named DeepFEM for solving nonlinear partial differential equations (PDEs) in solid mechanics is developed to reduce the number of sampling points required for training the deep neural network. Shape functions are introduced into deep learning to approximate the displacement field within the element. A general scheme for training the deep neural network based on derivatives computed from the shape functions is proposed. For the sake of demonstrations, the nonlinear vibration, nonlinear bending, and cohesive fracture problems are solved, and the results are compared with those from the existing methods to evaluate the performance of the present method. The results demonstrate that DeepFEM can effectively approximate the solution of the nonlinear mechanics problems with high accuracy, while the shape functions can significantly improve the computational efficiency. Moreover, with the trained DeepFEM model, the solutions of nonlinear problems with different geometric or material properties can be obtained instantly without retraining. Finally, the proposed DeepFEM is employed in the identification of material parameters of composite plate. The results show that the longitudinal and transverse elastic moduli of the ply in the composite plates can be accurately predicted based on the nonlinear mechanical response of plates.

Perturbed‐analytic direct transcription for optimal control (PADOC)

AbstractThe herein coined PADOC (perturbed‐analytic direct transcription for optimal control) stands for a new transcription method for direct trajectory optimization. We construct PADOC on a novel segmented decomposition method that provides series solutions for nonlinear problems. To transcribe the infinite dimensional problem into a finite one, PADOC comes along with a new solution approach, namely, the herein coined average nonlinear programming (aNLP). The aNLP theorem suggests generating a staircase optimal solution (low resolution) and turning it into a distributed optimal solution (high resolution) by exploiting the Hamiltonian of the problem. The analytic architecture provides PADOC with an analytic connection of a truncated order between the discrete nodes of the solution. This renders stability, accuracy within an analytic resolution, robustness, and a cut‐down in the number of the decision variables in the frame of NLP. We also prove that the multipliers associated with the Karush–Kuhn–Tucker optimality conditions in the frame of NLP (as transcribed by PADOC) correspond to a backward analytic solution for the costate equations. We finally show distinct features of PADOC through some examples.

Numerical algorithms for solutions of nonlinear problems in some distance spaces

<abstract><p>This paper introduces some numerical algorithms for finding solutions of nonlinear problems like functional equations, split feasibility problems (SFPs) and variational inequality problems (VIPs) in the setting of Hilbert and Banach spaces. Our approach is based on the Thakur-Thakur-Postolache (TTP) iterative algorithm and the class of mean nonexpansive mappings. First we provide some convergence results (including weak and strong convergence) in the setting of Banach space. To support these results, we provide a numerical example and prove that our TTP algorithm in this case converges faster to fixed point compared to other iterative algorithms of the literature. After that, we consider two new TTP type projection iterative algorithms to solve SFPs and VIPs on the Hilbert space setting. Our result are new in analysis and suggest new type effective numerical algorithms for finding approximate solutions of some nonlinear problems.</p></abstract>

Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

<p style='text-indent:20px;'>In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water flow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy <inline-formula><tex-math id="M1">\begin{document}$ H = O(\sqrt{h}) $\end{document}</tex-math></inline-formula>.</p>

Convergence rate for the hybrid iterative technique to explore the real root of nonlinear problems

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.

Ion-acoustic wave dynamics and sensitivity study in a magnetized Auroral phase plasma

The dynamic behavior of ion-acoustic waves (IAWs) inside a magnetized auroral zone plasma including ions as well as two-temperature electrons is investigated. The Kadomtsev–Petviashvili (KP) and modified KP (mKP) equations are provided using the reductive perturbation method (RPT). In fluid dynamics, plasma physics, and electrodynamics, mKP equation can be utilized to characterize wave phenomena. It increases our interest due to its significant physical and mechanical value. The new extended direct algebraic approach is used to compute several traveling wave patterns for the nonlinear mKP equation. It is a simple, concise and general method in symbolic computations for solitary wave solutions for nonlinear problems. It yields dark solitary solutions, the community for semi-dark soliton results, dark singular soliton effects, kink soliton results, and soliton consequences encompassing trigonometric function, mixture hyperbolic function, as well as rational functions. Some arbitrarily chosen solutions are displayed visually for physical understanding. Moreover, the sensitivity of the perturbed dynamical structure equivalent to the mKP equation has been explored. Furthermore, the effect of IAWs spectral index on the parameters of the solitary-wave solutions and the sensitivity of the perturbed dynamical system of mKP equations is investigated in detail.

On a new nonlinear integro-differential Fredholm-Chandrasekhar equation

This paper presents an analytical and numerical study of a new integro-differential Fredholm-Chandrasekhar equation of the second type. We suggest the conditions that ensure the existence and uniqueness of the nonlinear problem's solution. Then, we create a numerical technique based on the Nystr\"{o}m's method. The numerical application illustrates the efficiency of the proposed process.

An Effective Hybrid Symbolic Regression–Deep Multilayer Perceptron Technique for PV Power Forecasting

The integration of Photovoltaic (PV) systems requires the implementation of potential PV power forecasting techniques to deal with the high intermittency of weather parameters. In the PV power prediction process, Genetic Programming (GP) based on the Symbolic Regression (SR) model has a widespread deployment since it provides an effective solution for nonlinear problems. However, during the training process, SR models might miss optimal solutions due to the large search space for the leaf generations. This paper proposes a novel hybrid model that combines SR and Deep Multi-Layer Perceptron (MLP) for one-month-ahead PV power forecasting. A case study analysis using a real Australian weather dataset was conducted, where the employed input features were the solar irradiation and the historical PV power data. The main contribution of the proposed hybrid SR-MLP algorithm are as follows: (1) The training speed was significantly improved by eliminating unimportant inputs during the feature selection process performed by the Extreme Boosting and Elastic Net techniques; (2) The hyperparameters were preserved throughout the training and testing phases; (3) The proposed hybrid model made use of a reduced number of layers and neurons while guaranteeing a high forecasting accuracy; (4) The number of iterations due to the use of SR was reduced. The presented simulation results demonstrate the higher forecasting accuracy (reductions of more than 20% for Root Mean Square Error (RMSE) and 30 % for Mean Absolute Error (MAE) in addition to an improvement in the R2 evaluation metric) and robustness (preventing the SR from converging to local minima with the help of the ANN branch) of the proposed SR-MLP model as compared to individual SR and MLP models.

Parametric analysis and a predictive model for color difference during laser-induced coloration on titanium.

Laser-induced coloration on a metallic surface has been of interest to many application arweas. However laser machining of metals involves many complex problems including nonlinear unstable coupled with multiple factors. Therefore there are still some significant challenges in the precise control of color creation. Here we explored the process of the laser-induced coloration and find the connection between surface colors and processing parameters. The Response Surface Methodology (RSM) based experimental design was adopted to explore the influence of the single processing parameter and the interaction between parameters on color changes of titanium. The results showed that the scanning speed laser power repetition rate and hatch distance had significant effects on color changes of titanium. Then we demonstrated that using artificial neural network (ANN) is an effective solution of nonlinear problems in laser-induced coloration which can match the processing parameters and the L*a*b* color values on titanium surface precisely with limited experiments. Finally we successfully used the processing parameters estimated by ANN model to create unique art painting on titanium with nanosecond pulsed laser. This work can provide a potential method to solve the problem in the color consistency and open a new perspective in industrial application of laser-induced coloration technology.

Scrutiny of principle of energy survival in beam-columns and analytical solution of governing partial nonlinear differential equation by new approach AYM

In this manuscript, we investigate and solve a complicated highly nonlinear differential equations in the beam-column in the Energy Survival Principal equation on the beam-column to show the ability of the first invented method which we named it AYM (Akbari Yasna’s Method). Certainly, we know that nonlinear beam-column process are very complex, and the governing differential equations governing them are nonlinearity complex. In this paper we present a new analytical solution which can easily analyze all such problems and make a great evolution in the nonlinear designing in the solid engineering (civil and mechanical and etc.). Finally, this scientific approach can create a great phenomenon in the analytical solution of nonlinear problems in engineering sciences, especially in the solids engineering.