Algorithm For Differential Equations Research Articles

Operational-matrix-based algorithm for differential equations of fractional order with Dirichlet boundary conditions

The fractional differential equations (FDEs) are ground-breaking tools to demonstrate the complex-nature scientific systems in the form of non-linear behavior endorsed by the scientific community to develop some new and accurate mathematical methods. The main objective of this paper is the development of an extended mathematical algorithm based on the Gegenbauer wavelet method for the fractional-order problem. The Gegenbauer wavelet operational matrix with their derivative is proposed in our study. Some new operational matrices for the derivative of fractional order with Dirichlet boundary condition is purposed by introducing the piecewise function. Furthermore, a successful use to analyze the solution for the set of algebraic equations governed through the extended Gegenbauer wavelets technique is performed. Analytical solutions of the mentioned problem are effectively obtained, and a comparative study is presented. The outcomes are obtained via the modified Gegenbauer wavelet method by endorsing the accuracy and effectiveness of the mentioned technique. The convergence and error bound analysis are enclosed in our investigation. It is further verified that the algorithm is quite accurate, and an efficient mathematical tool is used to tackle the nonlinear fractional-order complex-nature problems.

Forward‐in‐time goal‐oriented adaptivity

SummaryIn goal‐oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh to reduce the error of the solution in some quantity of interest. In time‐dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo‐dual problem that runs forward in time. We also describe a forward‐in‐time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one‐dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward‐in‐time dual problem once and then performs the adaptive process forwards in time.

Second-order algorithm for simulating stochastic differential equations with white noises

The second-order algorithm for simulating stochastic differential equations with Gaussian white noises is presented. These stochastic differential equations are universal type, among, these Gaussian white noises come from different sources. Specifically, the proposed algorithm extends previous first-order algorithm for stochastic differential equations with different white noises and second-order algorithm for stochastic differential equations with same white noise. In practice, it is proved that this algorithm is scientific.

Synchronization of Time-delay Chaotic Systems with Uncertainties and External Disturbances

In this article, the authors have studied the problem of synchronization of time-delay chaotic systems with uncertainties and external disturbances. The effectiveness of the problem statement is visualized through synchronization of time-delay advanced Lorenz system and double time-delay R¨ossler system with parametric uncertainties and disturbances using active control method. Numerical simulations are carried out using Runge-Kutta algorithm for delay differential equations (DDEs) and the results are depicted through graphs. The physical meaning of time-delay system is that a signal is transmitted and received at a later time, which is found to occur in active sensing problems. The salient feature of the article is the demonstration of the efficiency of the considered method during synchronization of time-delay chaotic systems even in presence of uncertainties and external disturbances.

Random bit multilevel algorithms for stochastic differential equations

We study the approximation of expectations E(f(X)) for solutions X of SDEs and functionals f:C([0,1],Rr)→R by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals f from the Lipschitz class w.r.t. the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms.

常微分方程常用的数值算法及其应用

常微分方程常用的数值算法及其应用

A New Parareal Algorithm for Problems with Discontinuous Sources

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary differential equations which involve discontinuous right-hand sides. Such situations occur in various applications, e.g., when an electric device is supplied with a pulse-width-modulated signal. Our new Parareal algorithm uses a smooth input for the coarse problem with reduced dynamics. We derive error estimates that show how the input reduction influences the overall convergence rate of the algorithm. We support our theoretical results by numerical experiments, and also test our new Parareal algorithm in an eddy current simulation of an induction machine.

Adaptive random Fourier features with Metropolis sampling

The supervised learning problem to determine a neural network approximation $ \mathbb{R}^d\ni x\mapsto\sum_{k = 1}^K\hat\beta_k e^{{{\mathrm{i}}}\omega_k\cdot x} $ with one hidden layer is studied as a random Fourier features algorithm. The Fourier features, i.e., the frequencies $ \omega_k\in\mathbb{R}^d $, are sampled using an adaptive Metropolis sampler. The Metropolis test accepts proposal frequencies $ \omega_k' $, having corresponding amplitudes $ \hat\beta_k' $, with the probability $ \min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^{\gamma}\big\} $, for a certain positive parameter $ {\gamma} $, determined by minimizing the approximation error for given computational work. This adaptive, non-parametric stochastic method leads asymptotically, as $ K\to\infty $, to equidistributed amplitudes $ |\hat\beta_k| $, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods. Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is tested both on synthetic data and a real-world high-dimensional benchmark.

A compact integrated RBF method for time fractional convection–diffusion–reaction equations

In this paper, a local compact integrated radial basis function (CIRBF) method is proposed to solve the time fractional convection–diffusion–reaction equations. The proposed CIRBF scheme derives high accuracy and fast convergence rate with grid refinement. The spatial derivatives are discretized by combining with the integrated RBF approximation and compact approximation in a stencil. The nodal function values and the nodal second derivatives are used to establish the relation between the physical space and the RBF weight space. The time discretization is replaced by the second order shifted Grünwald scheme. Moreover, the unconditional stability and convergence of the proposed method are shown. Finally, some numerical examples on 2D and 3D domains are carried out to verify the accuracy and efficiency of the proposed algorithm for the fractional differential equations.

Efficient high order algorithms for fractional integrals and fractional differential equations

We propose an efficient algorithm for the approximation of fractional integrals by using Runge–Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time.

Symplectic Radial Basis Approximation of Multi-variate Hamiltonian PDEs

A meshless symplectic algorithm for multi-dimensional linear Hamiltonian partial differential equations with radial basis interpolation has been suggested when the solving domain is $${\mathbb {R}}^d$$ . In this study, for sake of application, the energy error estimate of the proposed symplectic scheme is made when the targeted domain is bounded. The paper also extends the algorithm to solve the nonlinear two-dimensional Klein–Gordon equation. As numerical experiments shown, the algorithm preserves the discretization energy, and it is efficient to deal with the nonlinear case.

Volumetric variational principles for a class of partial differential equations defined on surfaces and curves

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace-Beltrami equations and surface wave equations. The approach is to systematically formulate extensions of the variational integrals, derive the Euler-Lagrange equations of the extended problem, including the boundary conditions} that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions, and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher order regularization term or by periodically but infrequently "reinitializing" the computed solutions. Some numerical examples for each representative surface PDEs are presented.

Water Hammer Peak Pressures and Decay Rates of Transients in Smooth Lines With Turbulent Flow

Transient pressure peak values and decay rates associated with water hammer surges in fluid lines are investigated using an analytical method that has been formulated, in a previous publication, to simulate pressure transients in turbulent flow. The method agrees quite well with method of characteristics (MOC) simulations of unsteady friction models and has been verified with experimental data available for Reynolds numbers out to 15,800. The method is based on the formulation of ordinary differential equations from the frequency response of a pressure transfer function using an inverse frequency algorithm. The model is formulated by dividing the line into n-sections to distribute the turbulence resistance along the line at higher Reynolds numbers. In this paper, it will be demonstrated that convergence of the analytical solution is achieved with as few as 5–10 line sections for Reynolds numbers up to 200,000. The method not only provides for the use of conventional time domain solution algorithms for ordinary differential equations but also provides empirical equations for estimating peak surge pressures and transient decay rates as defined by eigenvalues. For typical sets of line and fluid properties, the trend of the damping ratio of the first or dominate mode of the pressure transients transfer function is found to be an approximate linear function of a dimensionless parameter that is a function of the Reynolds number. In addition, a reasonably accurate dimensionless trend formula for estimates of the normalized peak pressures is formulated and presented.

An efficient numerical algorithm for multi-dimensional time dependent partial differential equations

An efficient and robust numerical scheme based on Haar wavelets and finite differences is suggested for the solution of two-dimensional time dependent linear and nonlinear partial differential equations (PDEs). Excellent feature of the scheme is the conversion of linear and non-linear PDEs to algebraic equations which are comparatively easy to handle. Convergence of the scheme, which guarantees small error norm as the resolution level increases, is also an important part of this work. Different error norms are computed to check efficiency of the technique. Computations verify accuracy, flexibility and low computational cost of the method.

Parareal algorithms with local time-integrators for time fractional differential equations

It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal method to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.

Symmetry Reduction of Ordinary Differential Equations Using Moving Frames

The symmetry reduction algorithm for ordinary differential equations due to Sophus Lie is revisited using the method of equivariant moving frames. Using the recurrence formulas provided by the theory of equivariant moving frames, computations are performed symbolically without relying on the coordinate expressions for the canonical variables and the differential invariants occurring in Lie’s original procedure.

A Fast and Spectrally Convergent Algorithm for Rational-Order Fractional Integral and Differential Equations

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ODEs [S. Olver and A. Townsend, SIAM Rev., 55 (2013), pp. 462--489] and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.

An Improved Numerical Algorithm for the Fractional Differential Equations and Its Application in the Fractional-Order Nonlinear Systems

In this paper, an improved numerical algorithm for the fractional differential equations is proposed based on the variational iteration method. Using the improved numerical scheme, we investigate a new fractional-order hyperchaotic system, and find that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4. The lowest order we find to yield hyperchaos is 3.46 in this new fractional-order system. The existence of two positive Lyapunov exponents further verifies our results. Numerical results show that the proposed method has a faster speed and more accurate comparing with the traditional predictor-corrector algorithm. Based on the stability theory of the fractional-order system, a nonlinear controller is constructed to achieve synchronization fora class of nonlinear fractional-order systems using the pole placement technique. The nonlinear control method can synchronize two different fractional-order hyperchaotic systems. This synchronization scheme which is simple and theoretically rigorous enables synchronization of fractional-order hyperchaotic systems to be achieved in a systematic way and does not need to compute the conditional Lyapunov exponents. Simulation results are given to validate the effectiveness of the proposed synchronization method.

The “No Sampling Parameter Estimation (NSPE)” algorithm for stochastic differential equations

The parameter estimation problem in stochastic differential equations (SDEs) has gained much attention in recent years due to their increased applications in the field of pharmaceuticals, ecosystem modeling, and medical data like EKG, blood pressure, and sugar levels. The predictive power of SDEs lie in the choice of parameter values that describe the real data effectively. The classical SDE parameter estimation methods are largely based on likelihood estimation, which is computationally expensive. In this work, we propose a relatively simplified approach based on deterministic nonlinear optimization method which does not require sampling. The results from the suggested No Sampling Parameter Estimation (NSPE) algorithm for selected examples are compared with the data, results from deterministic ODE (ordinary differential equation) model and traditional methods of maximum likelihood estimation (MLE) and generalized method of moments (GMM) for SDEs. The NSPE algorithm is more accurate and reduces computation time significantly when compared to the traditional methods.

An algorithm for partial functional differential equations modeling tumor growth

We introduce a parallel algorithm for the numerical simulation of the growth of human tumor cells in time-varying environments and their response to therapy. The behavior of the cell populations is described by a system of delay partial differential equations with time-dependent coefficients. We construct the new algorithm by developing a time-splitting technique in which the entire problem is split into independent tasks assigned to arbitrary numbers of processors chosen in light of available resources. We present the results of a series of numerical experiments, which confirm the efficiency of the algorithm and exhibit a substantial decrease in computational time thus providing an effective means for fast clinical, case-by-case applications of tumor invasion simulations and possible treatment.