Method For Solution Of Equations Research Articles

An Hermite-Obreschkoff method for stiff high-index DAE

We have developed an implicit Hermite-Obreschkoff method for the numerical solution of stiff high-index differential-algebraic equations (DAEs). On each integration step, this method requires the computation of Taylor coefficients and their gradients to construct and solve a nonlinear system for the numerical solution, which is then projected to satisfy the constraints of the problem. We derive this system, show how to compute its Jacobian through automatic differentiation, and present the ingredients of our method, such as predicting an initial guess for Newton’s method, error estimation, and stepsize and order control. We report numerical results on stiff DAEs illustrating the accuracy and performance of our method, and in particular, its ability to take large steps on stiff problems.

Right-hand side method for the numerical solution of nonlinear partial differential equations

Right-hand side method for the numerical solution of nonlinear partial differential equations

Incremental harmonic balance method for multi-harmonic solution of high-dimensional delay differential equations: Application to crossflow-induced nonlinear vibration of steam generator tubes

Diverse science and engineering problems are governed by delay differential equations (DDE). Seeking periodic solutions of DDEs is crucial for many nonlinear dynamic systems. The incremental harmonic balance (IHB) method is an efficient semi-analytical approach for periodic solutions of DDE. Among various DDE systems, flow-induced vibration (FIV) of tube bundles with loose support is unique, in the sense that the added damping cancels out structural damping in the vicinity of critical velocity, making it extremely slow to retain a convergent solution if numerical integration (NI) is utilized. Despite the efforts of developing IHB for various mechanical vibration problems with time-delay, no attempt has been made to employ IHB to nonlinear FIV problems. Here we fill this blank by separately combining two spatial discretization strategies, i.e., discretization via linear vibration modes or via finite element method (FEM), with IHB to capture limit cycle solutions after the onset of instability. Within the range of gap velocity of interest, the IHB demonstrates excellent convergence by increasing harmonic terms, and good agreement is obtained between IHB results and the results by NI method. An 18 degree-of-freedom (DOF) nonlinear DDE system was readily dealt by integrating FEM and IHB, each DOF being approximated by three harmonics. This is the first attempt to employ IHB for multi-harmonic solution of high-dimensional FIV problems, which opens a door for solutions of other DDEs. All the codes used in this paper could be downloaded via: https://github.com/XJTU-Zhou-group/IHB.

Numerical Approximation for Direct Solution of Third-Order Ordinary Differential Equations Using New Class of Falkner-Type Block Method

In this article, a new class of Falkner-type methods for direct solution of third order ordinary differential equation is developed. The approach of collocation and interpolation technique is adopted to derive the new Falkner-type methods, which is implemented in block mode to get approximation at grids points simultaneously. The resulting scheme is zero-stable, consistent and convergent with good region of absolute stability. The tabular and graphical presentations of the numerical results to the problems considered, demonstrate the effectiveness and good accuracy of the scheme in comparison with other methods

Spectral Quasilinearization Method for the Numerical Solution of the Non-standard Volterra Integral Equations

In this paper, an iterative scheme is proposed through quasilinearization and the Legendre spectral collocation method to approximate the solution of the non-standard Volterra integral equations (NVIEs). At first, the process of quasilinearization serves to turn a NVIE into a sequence of linear Volterra integral equations. Then, in each iteration, the linear problem is solved by the Legendre spectral collocation method. After that, a convergence analysis is performed. The efficiency and accuracy of the proposed scheme are also verified through several illustrative examples.

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

A continuous five-step implicit block unification method for numerical solution of second-order elliptic partial differential equations

A continuous five-step implicit block unification method for numerical solution of second-order elliptic partial differential equations

Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

Stability domains of an implicit method for the numerical solution of Abel type integral algebraic equations

Stability domains of an implicit method for the numerical solution of Abel type integral algebraic equations

On the Reliability of Kolgan Reconstruction in Godunov Method for the Numerical Solution of Hydrodynamic Equations

On the Reliability of Kolgan Reconstruction in Godunov Method for the Numerical Solution of Hydrodynamic Equations

Deep learning for gradient flows using the Brezis–Ekeland principle

We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven.

A Comparative Survey of an Approximate Solution Method for Stochastic Delay Differential Equations

This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown.

A Solution Method for Differential Equations Based on Taylor PINN

A Solution Method for Differential Equations Based on Taylor PINN

A Novel Two Point Optimal Derivative free Method for Numerical Solution of Nonlinear Algebraic, Transcendental Equations and Application Problems using Weight Function

It’s a big challenge for researchers to locate the root of nonlinear equations with minimum cost, lot of methods are already exist in literature to find root but their cost are very high In this regard we introduce a two-step fourth order method by using weight function. And proposed method is optimal and derivative free for solution of nonlinear algebraic and transcendental and application problems. MATLAB, Mathematica and Maple software are used to solve the convergence and numerical problems of proposed and their counterpart methods.

New Analytical Method For Solution of Second Order Ordinary Differential Equations With Variable Coefficients

In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. In the first case, the method directly gives the solution in explicit or integral form. In the second method, the solution of the problem reduces to the solutions of adjoined second order ordinary differential equation of homogeneous type. As long as the analytical solution of two adjoined equations can be solved, the analytical solution can always be found. In the third method, the differential equation is transformed into Riccati equation. Riccati equation is solved by means of a method recently developed. In order to solve non-homogeneous differential equation, the fourth method is developed. The strategy is different but the solution is again based on the solution of Riccati equation.

The Absorption Properties of One‐Dimensional Spherical Photonic Crystals Based on Magnetized Ferrite Materials

AbstractBased on a spherical coordinate system and Maxwell's set of equations, the calculation formulas of one‐dimensional (1‐D) magnetized ferrite spherical photonic crystals (MFSPCs) are obtained and treated with the solution method of the spherical Bessel equations. Combined with the conventional transmission matrix method, expressions for the absorptance, reflectance, and transmittance of the 1‐D MFSPCs are given. 1‐D MFSPCs consisting of the general dielectric and Yttrium iron garnet (YIG) are also designed to verify the correctness of the conclusions. Variables such as external magnetic field, incidence angle, initial radius, and YIG thickness effects on the features of such 1‐D MFSPCs are also discussed. The transformation of the external magnetic field has a linear relationship with the position of the absorption band (AB). The three variables of incidence angle, initial radius, and YIG thickness have a substantial influence on the width and position of the AB, which can be adjusted according to the actual demand and are of high application significance. The results of this study can be helpful for the design of multifunctional absorbers and radome.

A symplectic homotopy perturbation method for random structural dynamics response analysis based on Birkhoffian systems

With the enhancement of the engineering structure complexity, numerical computation plays a prominent role in structural dynamics analysis. In theory, general mechanical systems can be transformed into Birkhoffian forms, whose matching symplectic numerical algorithms perform well in accuracy and stability. Besides, the influence of uncertainty cannot be ignored in practical engineering, but the research on uncertainty symplectic algorithms of Birkhoffian systems is still weak. This paper proposes a symplectic homotopy perturbation method (SHPM) for general Birkhoffian equations with random uncertainties. Different from the traditional perturbation method, the proposed method does not depend on small parameters. Based on the homotopy perturbation method, embedded parameters are introduced to establish the matrix and parameter perturbation equations of random Birkhoffian equations, respectively. Then the formulas for calculating the mean value and standard deviation of the response are derived with probability theory. Combined with Birkhoffian symplectic preserving algorithms, a symplectic homotopy perturbation method for the numerical solution of random Birkhoffian equations is proposed. Finally, numerical examples are given to verify the applicability and effectiveness of the proposed method and its superiority compared with the traditional non-symplectic method.

The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques

• The INTERNODES method is applied to problems in contact mechanics. • We propose a highly efficient preconditioner to solve the sequence of linear systems. • An algorithm is suggested to cheaply solve linear systems with the preconditioning matrix. • The quality of the preconditioner is assessed on small to medium size 2D problems. The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems.

A Controllability Method for Maxwell's Equations

We propose a controllability method for the numerical solution of time-harmonic Maxwell's equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a nonintrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap postprocessing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence toward the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.

An efficient, robust and reliable initialization of phase fractions in Rachford-Rice system of equations for three-phase split calculation

Multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three phases may co-exist. Solution of the Richford-Rice system of equations is an important operation in the multiphase flash. Within the literature, solution methods have been proposed, based on optimization methods and on nonlinear equation solution methods. Many methods in these two approaches may fail if the initial estimate is too inaccurate, and may require a substantial amount of computation only to arrive at a trivial solution in cases where the assumed number of phases is too high. The present work deals with an initialization method for the solution of the Rachford-Rice system of equations based on approximating each of the functions by means of tangent lines. It is proposed to solve the system of equations by applying a two-step Newton-type method. Through extensive numerical experimentation, the efficiency, robustness and reliability of the proposed initialization method for the solution of the Rachford-Rice system of equations by means of a Newton-type method is shown.