One-dimensional Nonlinear Equation Research Articles

Application of two-sided approximations method to solution of first boundary value problem for one-dimensional nonlinear heat conductivity equation

The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes.

Photovoltaic module efficiency evaluation: The case of Iraq

This study aims to evaluate the performance of a photovoltaic module under some extreme climate conditions, and with a case study for Iraq. CFD model is developed for the analysis of the photovoltaic module using the commercial CFD software of COMSOL Multiphysics v5.3a for the transient conditions. The results are verified with the analytical solution to the one-dimensional non-linear energy balance equation using Matlab. The results are also compared with measurements reported in the literature for validation. The results reveal that the free convection currents in inclined and horizontal positions of the module were weaker relative to the vertical position. Also, the increase in the length of inclined photovoltaic module, up to 1.3 m, enhances the heat transfer rate. However, beyond this length, the temperature of the module becomes higher, and the convective heat transfer coefficients are reduced regardless of the inclination. In the horizontal position, the convective heat transfer rate is lower, particularly on the bottom surface of PV system.

A Fast Preconditioned Semi-Implicit Difference Scheme for Strongly Nonlinear Space-Fractional Diffusion Equations

In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respectively. Stability and convergence properties of the proposed scheme are proved with the help of a discrete Grönwall inequality. Moreover, we extend the method to the solution of two-dimensional nonlinear models. A fast matrix-free implementation based on preconditioned Krylov subspace methods is presented for solving the discretized linear systems. The resulting fast preconditioned semi-implicit difference scheme reduces the memory requirement of conventional semi-implicit difference schemes from O(Ns2) to O(Ns) and the computational complexity from O(Ns3) to O(NslogNs) in each iterative step, where Ns is the number of space grid points. Experiments with two numerical examples are shown to support the theoretical findings and to illustrate the efficiency of our proposed method.

Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations

We consider a process (Xt) solution of a one-dimensional nonlinear self-stabilizing stochastic differential equation, with classical drift term V(α,x) depending on an unknown parameter α, self-stabilizing term Φ(β,x) depending on another unknown parameter β and small noise amplitude ɛ. Self-Stabilization is the effect of including a mean-field interaction in addition to the state-dependent drift. Adding this term leads to a nonlinear or McKean-Vlasov Markov process with transitions depending on the distribution of Xt. We study the probabilistic properties of (Xt) as ɛ tends to 0 and exhibit a Gaussian approximating process for (Xt). Next, we study the estimation of (α,β) from a continuous observation of (Xt,t∈[0,T]). We build explicit estimators using an approximate log-likelihood function obtained from the exact log-likelihood function of a proxi-model. We prove that, for fixed T, as ɛ tends to 0, α can be consistently estimated with rate ɛ−1but notβ. Then, considering ni.i.d. sample paths (Xti,i=1,…,n), we build consistent and asymptotically Gaussian estimators of (α,β) with rates nɛ−1 for α and n for β. Finally, we prove that the statistical experiments generated by (Xt) and the proxi-model are asymptotically equivalent in the sense of the Le Cam Δ-distance both for the continuous observation of one path and for ni.i.d. paths under the condition nɛ→0, which justifies our statistical method.

Energy-conserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions

In this article, we discuss the numerical analysis for the finite difference scheme of the one-dimensional nonlinear wave equations with dynamic boundary conditions. From the viewpoint of the discrete variational derivative method we propose the derivation of the energy-conserving finite difference schemes of the problem, which covers a variety of equations as widely as possible. Next, we focus our attention on the semilinear wave equation, and show the existence and uniqueness of the solution for the scheme and error estimates with the help of the inherited energy structure.

APPLICATION OF THE FICTITIOUS DOMAIN METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

The article shows the ways of applying the method of fictitious domains in solving problems for ordinary differential equations. In the introduction, a small review of the literature on this method, as well as methods for the numerical solution of these problems, is made. The problem statement for the method of fictitious domains for ordinary differential equations is considered. Further, the inequality of estimates was shown. The solution of the auxiliary problem approximates the solution of the original problem with a certain accuracy. The inequality of estimates is obtained in the class of generalized solutions. For the purpose of visual application of the fictitious domain method in problems, a boundary value problem for a one-dimensional nonlinear ordinary differential equation is considered. The problem was written in the form of a difference scheme and led to a solution using the sweep method. In the numerical solution of the problem, numerical calculations were carried out for various values of the parameter included in the auxiliary problem, based on the method of fictitious domains. The numbers of iterations, execution time, and graphs of these calculations are presented and analyzed.

Some engineering applications of newly constructed algorithms for one-dimensional non-linear equations and their fractal behavior

The aim of this paper is to develop some novel numerical algorithms for finding roots of one-dimensional non-linear equations. We derive these algorithms by utilizing the main and basic idea of the variational iteration technique. The convergence rate of the suggested algorithms is discussed. It is corroborated that the proposed numerical algorithms possess sixth-order convergence. To demonstrate the validity, applicability, and the performance of the proposed algorithms, we solved different test problems. These problems also include some real-life applications associated with the chemical engineering such as van der Wall’s equation, conversion of nitrogen-hydrogen feed to ammonia and the fractional-transformation in the chemical reactor problem. The numerical results of these problems show that the proposed algorithms are more effective against the other well-known similar nature existing methods. Finally, the dynamics of the suggested algorithms in the form of the polynomiographs of different complex polynomials have been analyzed that reveals the fractal nature and the other dynamical aspects of the suggested algorithms.

A Novel Rekasius Substitution Based Exact Method for Delay Margin Analysis of Multi-Area Load Frequency Control Systems

Time delays are inevitable in load frequency control (LFC) of the power system, especially in a deregulated system where open communication networks are more favorable than the dedicated ones. However, extensive delays may affect the stability of a system. In this paper, an exact frequency-domain method for calculating the delay margin of LFC systems is presented, which is likely to be used to assess a system's stability caused by time-delays and guide the controller design. Firstly, a dynamic model of load frequency control systems is established, described by delay differential and algebraic equations (DDAEs). The Rekasius substitution is then employed to eliminate the transcendental terms in the characteristic equation without making any approximation. Afterward, an eigenvalue perturbation theory is introduced to transform the problem of calculating the delay margin into solving a series of one-dimensional nonlinear equations. Since only equivalent transforms are involved in the whole process, the delay margins obtained by the proposed method are exact. Finally, a two-area and a three-area deregulated LFC systems are applied to verify the correctness and efficiency of the proposed method.

Numerical solution of nonlinear advection diffusion reaction equation using high-order compact difference method

In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of the first-order derivative, and the time derivative term is discretized by using the fourth-order backward differencing formula. An unconditionally stable five-step fourth-order fully implicit compact difference scheme is developed. This scheme has fourth-order accuracy in both time and space. Secondly, for the calculations of the start-up time steps, the time derivative term is discretized by the Crank-Nicolson method, and Richardson extrapolation formula is used to improve the accuracy in time direction from the second-order to the fourth-order. Thirdly, convergence and stability of the difference scheme in H1 seminorm, L∞ and L2 norms, existence and uniqueness of the numerical solutions are proved, respectively. Fourthly, the Thomas algorithm is used to solve the nonlinear algebraic equations at each time step, and a time advancement algorithm with linearized iteration strategy is established. Finally, the accuracy, stability and efficiency of the present approach are verified by some numerical experiments.

Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z-graded graphs

We consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.

Accurate and efficient approximations for generalized population balances incorporating coagulation and fragmentation

This study focuses on development of two approaches based on finite volume schemes for solving both one-dimensional and multidimensional nonlinear simultaneous coagulation-fragmentation population balance equations (PBEs). Existing finite volume schemes and sectional methods such as fixed pivot technique and cell average technique have many issues related to accuracy and efficiency. To resolve these challenges, two finite volume schemes are developed and compared with the cell average technique along with the exact solutions. The new schemes have features such as simpler mathematical formulations, easy to code and robust to apply on nonuniform grids. The numerical testing shows that both new finite volume schemes compute the number density functions and their corresponding integral moments with higher precision on a coarse grid by consuming lesser CPU time. In addition, both schemes are extended to approximate generalized simultaneous coagulation-fragmentation problems and retains the numerical accuracy and efficiency. For the higher dimensional PBEs (2D and 3D), the investigation and verification of the numerical schemes is done by deriving new exact integral moments for various combinations of coagulation kernels, selection functions and fragmentation kernels.

Global Existence for Two Singular One-Dimensional Nonlinear Viscoelastic Equations with respect to Distributed Delay Term

In this current work, we are interested in a system of two singular one-dimensional nonlinear equations with a viscoelastic, general source and distributed delay terms. The existence of a global solution is established by the theory of potential well, and by using the energy method with the function of Lyapunov, we prove the general decay result of our system.

Modelling and simulation of a wave energy converter

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use the characteristic equations of Riemann invariants to obtain the discretized transmission conditions and we implement the Lax-Friedrichs scheme to get numerical solutions.

Variational assimilation of surface wave data for bathymetry reconstruction. Part I: algorithm and test cases

Accurate mapping of ocean bathymetry is needed for effective modelling of ocean dynamics, such as tsunami prediction. Available bathymetry data does not always provide the resolution to model such nonlinear waves accurately, and collection of accurate data is logistically challenging. As an alternative, in this study we develop and evaluate a variational data assimilation scheme for the one-dimensional nonlinear shallow water equations that estimates bathymetry using a finite set of observations of surface wave height. We demonstrate that convergence to exact bathymetry is improved by including more observation locations and by implementing a low-pass filter in the data assimilation algorithm to remove small-scale noise. A necessary condition for convergence of the bathymetry reconstruction is that the amplitude of the initial conditions is less than 1% of the bathymetry height. We use density-based global sensitivity analysis (GSA) to assess the sensitivity of the surface wave and reconstruction error to model parameters. By demonstrating low sensitivity of the surface wave to the reconstruction error, we show that reconstructing the bathymetry with a relative error of about 10% is sufficiently accurate for surface wave modelling in most cases. These results can be used to guide the development of similar assimilation schemes in higher dimensions and more realistic geometries.

Some New Iterative Algorithms for Solving One-Dimensional Non-Linear Equations and Their Graphical Representation

Solving non-linear equation is perhaps one of the most difficult problems in all of numerical computations, especially in a diverse range of engineering applications. The convergence and performance characteristics can be highly sensitive to the initial guess of the solution for most numerical methods such as Newton's method. However, it is very difficult to select reasonable initial guess of the solution for most systems of non-linear equations. Besides, the computational efficiency is not high enough. Taking this into account, based on variational iteration technique, we develop some new iterative algorithms for solving one-dimensional non-linear equations. The convergence criteria of these iterative algorithms has also been discussed. The superiority of the proposed iterative algorithms is illustrated by solving some test examples and comparing them with other well-known existing iterative algorithms in literature. In the end, the graphical comparison of the proposed iterative algorithms with other well-known iterative algorithms have been made by means of polynomiographs of different complex polynomials which reflect the fractal behavior and dynamical aspects of the proposed iterative algorithms.

Conditional stability of multi-solitons for the 1D NLKG equation with double power nonlinearity

We consider the one-dimensional nonlinear Klein-Gordon equation with a double power focusing-defocusing nonlinearity∂t2u−∂x2u+u−|u|p−1u+|u|q−1u=0,on[0,∞)×R, where 1<q<p<∞. The main result states the stability in the energy space H1(R)×L2(R) of the sums of decoupled solitary waves with different speeds, up to the natural instabilities. The proof is inspired by the techniques developed for the generalized Korteweg-de Vries equation and the nonlinear Schrödinger equation in a similar context by Martel, Merle and Tsai [14,15]. However, the adaptation of this strategy to a wave-type equation requires the introduction of a new energy functional adapted to the Lorentz transform.

Ultradiscrete bifurcations for one dimensional dynamical systems

Bifurcations of one dimensional dynamical systems are discussed based on some ultradiscrete equations. The ultradiscrete equations are derived from normal forms of one-dimensional nonlinear differential equations, each of which has saddle-node, transcritical, or supercritical pitchfork bifurcations. An additional bifurcation, which is similar to the flip bifurcation, is found in ultradiscrete equations for supercritical pitchfork bifurcations. Dynamical properties of these ultradiscrete bifurcations can be characterized with graphical analysis. As an example of application of our treatment, we focus on an ultradiscrete equation of the FitzHugh–Nagumo model and discuss its dynamical properties.

Refined Nonlinear Deformation Model of Sandwich Plates with Composite Facings and Transversal-Soft Core

Following up on the results obtained earlier, a refined nonlinear model of static deformation of sandwich plates with transversal-soft core and facings with low stiffness of transverse shear and transverse compression is constructed for the case of cylindrical bending. It is based on the use of linear approximations in thickness for deflections of the external layers, cubic approximation in thickness for tangential displacements, and simplified three-dimensional equations of elasticity theory that can be integrated along the transverse coordinate with introduction of two unknown functions representing constant transverse tangential stresses in thickness for transversal-soft core. The kinematic relations for the facings are constructed in a geometrically nonlinear quadratic approximation. They allow, taking into account physical nonlinear behavior of the material under transverse shear conditions, the description of non-classical transverse-shear forms of stability loss (FSL) in both compression and bending conditions. Based on the generalized Lagrange variational principle, one-dimensional nonlinear equations of equilibrium and conjugation of the facings with the core by tangential displacements are constructed to describe the static deformation process with high rates of variability of the parameters of the stress-strain state (SSS).

Short-wavelength soliton in a fully degenerate quantum plasma

We present a novel one-dimensional nonlinear evolution equation governing the dynamics short-wavelength longitudinal waves in a nonrelativistic fully degenerate quantum plasma using kinetic equation for the Wigner function. The linear dispersion of the equation has a form of “zero sound” ∼k exp (−k2), where k is the wave number, and it strongly differs from previously known nonlinear evolution equations. We numerically find the corresponding soliton solutions and demonstrate that the collisions between three solitons turn out to be elastic, resulting only in phase shifts.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

&lt;p style='text-indent:20px;'&gt;In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [&lt;xref ref-type="bibr" rid="b21"&gt;21&lt;/xref&gt;], with a high order RKEI method [&lt;xref ref-type="bibr" rid="b7"&gt;7&lt;/xref&gt;], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.&lt;/p&gt;