System Of Nonlinear Algebraic Equations Research Articles

High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations

AbstractThis article develops high‐order implicit time‐stepping methods combined with the fourth‐order central essentially‐non‐oscillatory (CENO) scheme for stiff three‐dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three‐dimensional multiblock body‐fitted grids with hexahedral cells. Several implicit time integration methods of third‐ and fourth‐order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge‐Kutta (ESDIRK4), and Rosenbrock‐type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian‐free inexact Newton–Krylov method with additive Schwarz preconditioning using block‐based incomplete LU decomposition. The performance of the high‐order implicit time‐stepping methods on smooth and stiff problems is compared with a standard fourth‐order explicit Runge‐Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth‐order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth‐order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A‐stable ESDIRK4 scheme proved to be the more robust and accurate implicit time‐marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high‐order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high‐order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high‐order accuracy, and it also represents the first to establish high‐order implicit time integration for high‐order CENO in space.

Mathematical Model and Characteristics of the Induction Motor with a Power Supply from a Current Source

. Methods and mathematical models for studying the modes and characteristics of the three-phase squirrel-cage induction motor with the power supplied to the stator winding from the current source have been developed. The specific features of the algorithms for calculating transients, steady-state modes and static characteristics are discussed. The results of the calculation of the processes and characteristics of induction motors with the power supply from the current source and the voltage source are compared. Steady-state and dynamic modes cannot be studied with a sufficient adequacy based on the known equivalent circuits; this requires using dynamic parameters, which are the elements of the Jacobi matrix of the system of equations of the electromechanical equilibrium. In the mathematical model, the state equations of the stator and rotor circuits are written in the fixed two-phase coordinate system. The transients are described by the system of differential equations of electrical equilibrium of the transformed circuits of the motor and the equation of the rotor motion and the steady-state modes by the system of algebraic equation. The developed algorithms are based on the mathematical model of the motor in which the magnetic path saturation and skin effect in the squirrel-cage bars are taken into consideration. The magnetic path saturation is accounted for by using the real characteristics of magnetizing by the main magnetic flux and leakage fluxes of the stator and rotor windings. Based on them, the differential inductances are calculated, which are the elements of the Jacobi matrix of the system of equations describing the dynamic modes and static characteristic. In order to take into account the skin effect in the squirrel-cage rotor, each bar along with the squirrel-cage rings is divided height-wise into several elements. As a result, the mathematical model considers the equivalent circuits of the rotor with different parameters which are connected by mutual inductance. The non-linear system of algebraic equations of electrical equilibrium describing the steady-state modes is solved by the parameter continuation method. To calculate the static characteristics, the differential method combined with the Newton’s Iterative refinement is used.

Non-polynomial B-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers\u2019 equations numerically

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval [0, L]. The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

Operational matrices based on the shifted fifth-kind Chebyshev polynomials for solving nonlinear variable order integro-differential equations

In this research, we study a general class of variable order integro-differential equations (VO-IDEs). We propose a numerical scheme based on the shifted fifth-kind Chebyshev polynomials (SFKCPs). First, in this scheme, we expand the unknown function and its derivatives in terms of the SFKCPs. To carry out the proposed scheme, we calculate the operational matrices depending on the SFKCPs to find an approximate solution of the original problem. These matrices, together with the collocation points, are used to transform the original problem to form a system of linear or nonlinear algebraic equations. We discuss the convergence of the method and then give an estimation of the error. We end by solving numerical tests, which show the high accuracy of our results.

Pipe roughness identification of water distribution networks: A Tensor method

The identification of pipe roughnesses in a water distribution network is formulated as a nonlinear system of algebraic equations which turns out to be demanding to solve under real-world circumstances. This paper proposes an enhanced technique to numerically solve this identification problem, extending the conventional Newton–Raphson approach with second-order derivatives in the determination of the search direction. Despite the requirement to solve a nonlinear equation to obtain a search direction, the application of the Hadamard/Schur product operator enables the resulting formulation to be represented compactly and thus facilitates the development of an efficient and more robust solving-technique. Algorithms on the basis of this more enhanced solving method are then compared to a customized Newton–Raphson approach in simulation examples.

Radial basis functions method for nonlinear time- and space-fractional Fokker-Planck equation

In this study, a radial basis functions (RBFs) method for solving nonlinear timeand space-fractional Fokker-Planck equation is presented. The time-fractional derivative is of the Caputo type, and the space-fractional derivatives are considered in the sense of Caputo or Riemann-Liouville. The Caputo and Riemann-Liouville fractional derivatives of RBFs are computed and utilized for approximating the spatial fractional derivatives of the unknown function. Also, in each time step, the time-fractional derivative is approximated by the high order formulas introduced in [6], and then a collocation method is applied. The centers of RBFs are chosen as suitable collocation points. Thus, in each time step, the computations of fractional Fokker-Planck equation are reduced to a nonlinear system of algebraic equations. Several numerical examples are included to demonstrate the applicability, accuracy, and stability of the method. Numerical experiments show that the experimental order of convergence is 4 − α where α is the order of time derivative.

Discrete collocation method for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations

This paper proposes a discrete collocation method based on the fuzzy Lagrange interpolation and the fuzzy Gauss-Legendre quadrature formula for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations. Firstly, the existence and uniqueness of the solution of the original equation are proved by using a two-dimensional Gronwall inequality and an iterative method. Secondly, the discrete collocation method is utilized to convert the linear and nonlinear fuzzy Volterra integral equations into the corresponding linear and nonlinear systems of algebraic equations. Finally, the error estimation and convergence analysis of the proposed numerical method are discussed in terms of the uniform modulus of continuity. Some numerical experiments illustrate that the scheme possesses a higher accuracy and yields a better approximation when compared with other known methods.

On the Use of Composite Functions in the Simple Equations Method to Obtain Exact Solutions of Nonlinear Differential Equations

We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of a class of nonlinear differential equations containing polynomial nonlinearities. We present an amended version of the methodology, which is based on the use of composite functions. The number of steps of the SEsM was reduced from seven to four in the amended version of the methodology. For the case of nonlinear differential equations with polynomial nonlinearities, SEsM can reduce the solved equations to a system of nonlinear algebraic equations. Each nontrivial solution of this algebraic system leads to an exact solution of the solved nonlinear differential equations. We prove the theorems and present examples for the use of composite functions in the methodology of the SEsM for the following three kinds of composite functions: (i) a composite function of one function of one independent variable; (ii) a composite function of two functions of two independent variables; (iii) a composite function of three functions of two independent variables.

Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional order integro-differential equation

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [−1,1] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

Fractional Chebyshev cardinal wavelets: application for fractional quadratic integro-differential equations

This paper introduces a new set of the basis functions called the fractional Chebyshev cardinal wavelets and details their properties. These wavelets have a greater degree of freedom than the classical Chebyshev cardinal wavelets. Moreover, they retain the cardinality and the spectral accuracy of these wavelets. The fractional derivative and integral matrices of these fractional basis functions are obtained exactly in the explicit forms. In this study, we aim to devise an efficient and powerful approximation method using these new basis functions. Then, we employ them for a new category of nonlinear fractional quadratic integro-differential equations. By employing their fractional integral matrix and their cardinality, the problem under study is transformed into solving a nonlinear system of algebraic equations. The error analysis of the presented technique is first investigated theoretically and then computational efficiency is examined for two numerical examples.

Laguerre polynomial-based operational matrix of integration for solving fractional differential equations with non-singular kernel

The Atangana–Baleanu derivative and the Laguerre polynomial are used in this analysis to define a new computational technique for solving fractional differential equations. To serve this purpose, we have derived the operational matrices of fractional integration and fractional integro-differentiation via Laguerre polynomials. Using the derived operational matrices and collocation points, we reduce the fractional differential equations to a system of linear or nonlinear algebraic equations. For the error of the operational matrix of the fractional integration, an error bound is derived. To illustrate the accuracy and the reliability of the projected algorithm, numerical simulation is presented, and the nature of attained results is captured in diverse order. Finally, the achieved consequences enlighten that the solutions obtained by the proposed scheme give better convergence to the actual solution than the results available in the literature.

Crank-Nicolson/finite element approximation for the Schrödinger equation in the de Sitter spacetime

Central to much of science, engineering, and society today is the building of mathematical models to represent complex processes. Recently, the non-relativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime has been used to derive Schrödinger equations with weighted nonlinear terms In this paper, numerical simulations are constructed to clarify the behavior of the solution in both one- and two-dimensions. These simulations are constructed based on the Crank-Nicolson scheme in the time direction and the Galerkin finite element in the spatial direction. The nonlinear system of algebraic equations resulting from the constructed scheme is solved using Newton’s method. It is also demonstrated that the numerical approximation converges to the exact one.

Numerical solution for a class of time-fractional stochastic delay differential equation with fractional Brownian motion

In this article, a numerical scheme is proposed to solve a class of time-fractional stochastic delay differential equations (TFSDDEs) with fractional Brownian motion (fBm). First, we convert the TFSDDE into a non-delay equation by using a step-by-step scheme. Then, by applying a collocation method based on Jacobi polynomials (JPs) in each step, the non-delay equation is reduced to a nonlinear system of algebraic equations. The convergence analysis of the presented scheme is evaluated. Finally, two numerical test examples are presented to highlight the applicability and efficiency of the investigated method.

Numerical simulations of MHD mixed convection of hybrid nanofluid flow in a horizontal channel with cavity: Impact on heat transfer and hydrodynamic forces

In this paper, we explore the characteristics of heat transfer and fluid flow of hybrid nanofluid enclosed in a horizontal channel with cavity having an obstacle. The dimensionless governing partial equations for various physical parameters are simulated via higher order and stable Galerkin based finite element method. In particular, the higher order and stable finite element pair, ¶2/¶1 is utilized for the discretization of the modelled partial differential equations. The discretised system of nonlinear algebraic equations are linearized with the help of Newton's method and the resulting linear problems are computed with the help of geometric multi-grid techniques. Effective governing parameters such as Reynolds number, cylinder rotation, aspect ratio of cavity and the impact of variance in cylinder radius on the heat transfer and fluid flow are analyzed in detail. Moreover, the computation of hydrodynamic forces such as drag and lift coefficients on the obstacle, are also analyzed in this paper. It is shown that by increasing the radius of the obstacle the heat transfer in the channel is enhanced up to 119%. Furthermore, horizontal position of cylinder only contributes a heat transfer efficacy of 2.54%.

A mathematical model and numerical solution for brain tumor derived using fractional operator

In this paper, we present a mathematical model of brain tumor. This model is an extension of a simple two-dimensional mathematical model of glioma growth and diffusion which is derived from fractional operator in terms of Caputo which is called the fractional Burgess equations (FBEs). To obtain a solution for this model, a numerical technique is presented which is based on operational matrix. First, we assume the solution of the problem under the study is as an expansion of the Bernoulli polynomials. Then with combination of the operational matrix based on the Bernoulli polynomials and collocation method, the problem under the study is changed to a system of nonlinear algebraic equations. Finally, the proposed technique is simulated and tested on three types of the FBEs to confirm the superiority and accuracy.

Two reliable methods for numerical solution of nonlinear stochastic Itô–Volterra integral equation

This article proposes two efficient methods to solve nonlinear stochastic Itô–Volterra integral equations. The shifted Jacobi spectral Galerkin method and shifted Jacobi operational matrix method have been applied to solve these equations. The presented methods convert this equation to a system of nonlinear algebraic equations and then Newton’s method have been implemented to solve the obtained algebraic equations numerically. Convergence analysis for both presented methods have been investigated. Also, the results obtained by proposed methods have been compared. The accuracy and reliability of the presented methods have been proved by some numerical instances.

Symplectic discrete-time energy-based control for nonlinear mechanical systems

We present a novel approach for discrete-time state feedback control implementation which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. We translate the argument of energy shaping to discrete time by using the symplectic implicit midpoint rule. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. For the nonlinear case, the sampled system and the target dynamics are approximated with second order accuracy using the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer–Verlet integration scheme on a staggered grid. Both the Hamiltonian and the Lagrangian perspective are adopted. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control in the simulation examples to illustrate the performance and stability gain compared to the quasi-continuous implementation. We discuss computational aspects and show the structural advantages of the implicit midpoint rule compared to other integration schemes in the appendix.

Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type

In this article, we develop a new set of functions called fractional-order Alpert multiwavelet functions to obtain the numerical solution of fractional pantograph differential equations (FPDEs). The fractional derivative of Caputo type is considered. Here we construct the Riemann–Liouville fractional operational matrix of integration (Riemann–Liouville FOMI) using the fractional-order Alpert multiwavelet functions. The most important feature behind the scheme using this technique is that the pantograph equation reduces to a system of linear or nonlinear algebraic equations. We perform the error analysis for the proposed technique. Illustrative examples are examined to demonstrate the important features of the new method.

Theoretical and computational analysis of nonlinear fractional integro-differential equations via collocation method

In this article, a class of nonlinear Volterra-Fredholm fractional integro-differential equations is considered, both theoretical and computational aspects. The respective theoretical results are devoted to the existence of a solution via fixed point approach. Further, for the computational aspect, the Proposed Methodology of Haar wavelet collocation. This method minimizes a system of nonlinear algebraic equations, which is developed by Broyden’s method. In literature, the proposed method is taken for checking the convergence with help of some numerical examples. Calculate mean square root and maximum absolute errors for various collocation point numbers. The final outcomes show that the applied Haar method is effective, and the convergence rate for different collocation point is roughly equal to 2.

Matrix computational collocation approach based on rational Chebyshev functions for nonlinear differential equations

In this work, a numerical technique for solving general nonlinear ordinary differential equations (ODEs) with variable coefficients and given conditions is introduced. The collocation method is used with rational Chebyshev (RC) functions as a matrix discretization to treat the nonlinear ODEs. Rational Chebyshev collocation (RCC) method is used to transform the problem to a system of nonlinear algebraic equations. The discussion of the order of convergence for RC functions is introduced. The proposed base is specified by its ability to deal with boundary conditions with independent variable that may tend to infinity with easy manner without divergence. The technique is tested and verified by two examples, then applied to four real life and applications models. Also, the comparison of our results with other methods is introduced to study the applicability and accuracy.