ODE Method Research Articles

A class of explicit second derivative general linear methods for non-stiff ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics

A Family of Conditionally Explicit Methods for Second-Order ODEs and DAEs: Application in Multibody Dynamics

Asymptotics for some logistic maps and the renormalization group

Abstract We explain the relation between the r = 1 case of the logistic map x i+1 = r x i (1 − x i ), x i ∈ R , i = 0, 1, 2, …, r > 0 and x 0 ≥ 0, and the renormalization group flow arising in the multiscale analysis of interesting zero fixed point, asymptotic free quantum field theory models such as the ultraviolet (1 + 1)-dimensional Gross-Neveu model and QCD, and the infrared ϕ 4 4 model . We obtain the asymptotics of the mapping, which shows an inverse power decay approach to the fixed point x * = 0, Gaussian fixed point, with additional logarithmic-like corrections. This asymptotic behavior is independent of the initial condition x 0 ∈ (0, 1) (hence, there is no constraint for x 0 to be small, as usual in quantum field models), and only depends on the lowest orders in a polynomial perturbation. In asymptotic free quantum field theory, this amounts to say that knowing the renormalization group β-function expansion in the coupling constant, up to higher orders, does not improve our knowledge of the asymptotics of the coupling flow. A comparison with a similar differential equation with continuous time is made by analyzing stability of this kind of solution and higher order monomial perturbations. We also obtain the detailed asymptotics for 0 < r < 1. As well, our methods can be applied when r ∈ (1, 3]. It is known, but without detailed asymptotics, that all trajectories with initial condition x 0 ∈ ( − 1, 1) converge to the fixed point x * = (r − 1)/r. For r = 2, the super attractive case, we obtain an explicit exact solution which exhibits an exploding, non-constant exponential decay rate approach to the x * = (1/2) fixed point. Our methods include the use of iterations, a discrete version of the Fundamental theorem of Calculus, a discrete version of the integrating factor method for first order linear ODEs and, sometimes, a scaling transformation. To obtain these results, we do not use the traditional Banach contraction mapping theorem, which only provides an upper bound on the asymptotics. We expect that our methods can be employed to determine the asymptotics of the logistic map for a wider range of parameters, where other fixed points are present.

SMS: Spiking marching scheme for efficient long time integration of differential equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy.

An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost

An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost

Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v, with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.

The sticky Lévy process as a solution to a time change equation

Stochastic Differential Equations (SDEs) were originally devised by Itô to provide a pathwise construction of diffusion processes. A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential Equations driven by random functions. We present a simple example where TCEs have some advantage over SDEs. We represent sticky Lévy processes as the unique solution to a TCE driven by a Lévy process with no negative jumps. The solution is adapted to the time-changed filtration of the Lévy process driving the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly. Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler's method for ODEs), whenever the driving Lévy process is strongly approximated.

Discontinuous Galerkin and Related Methods for ODE

Discontinuous Galerkin and Related Methods for ODE

Convergence of stochastic approximation via martingale and converse Lyapunov methods

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar and Meyn (SIAM J Control Optim 38:447–469, 2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a “converse” Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar and Meyn (2000).

A Filtered Milne-Simpson ODE Method with Enhanced Stability Properties

The Milne-Simpson method is a two-step implicit linear multistep method for the numerical solution of ODEs that obtains the theoretically highest order of convergence for such a method. The stability region of the method is only an interval on the imaginary axis and the method is classified as weakly stable which causes non-physical oscillations to appear in numerical solutions. For this reason, the method is seldom used in applications. This work examines filtering techniques that improve the stability properties of the Milne-Simpson method while retaining its fourth-order convergence rate. The resulting filtered Milne-Simpson method is attractive as a method of lines integrator of linear time-dependent partial differential equations.

Globally conformally Kähler Einstein metrics on certain holomorphic bundles

The subject of this paper is the explicit momentum construction of complete Einstein metrics by ODE methods. Using the Calabi ansatz, further generalized by Hwang-Singer, we show that there are non-trivial complete conformally Kähler–Einstein metrics on certain Hermitian holomorphic vector bundles and their subbundles over complete Kähler–Einstein manifolds. In special cases, we give the explicit expressions of these metrics. These examples show that there are a compact Kähler manifold M and its subvariety N whose codimension is greater than 1 such that there is a complete conformally Kähler–Einstein metric on M–N.

Least-Squares Spectral Methods for ODE Eigenvalue Problems

We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

The trace-reinforced ants process does not find shortest paths

In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $\mathcal G$. The ants perform a sequence of random walks on this graph, starting from the nest and stopped when first hitting the source of food. At each step of its random walk, the $n$-th ant chooses to cross a neighbouring edge with probability proportional to the number of preceding ants that crossed that edge at least once. We say that {\it the ants find the shortest path} if, almost surely as the number of ants grow to infinity, almost all the ants go from the nest to the source of food through one of the shortest paths, without loosing time on other edges of the graph. Our contribution is three-fold: (1) We prove that, if $\mathcal G$ is a tree rooted at $N$ whose leaves have been merged into node $F$, and with one edge between $N$ and $F$, then the ants indeed find the shortest path. (2) In contrast, we provide three examples of graphs on which the ants do not find the shortest path, suggesting that in this model and in most graphs, ants do not find the shortest path. (3) In all these cases, we show that the sequence of normalised edge-weights converge to a {\it deterministic} limit, despite a linear-reinforcement mechanism, and we conjecture that this is a general fact which is valid on all finite graphs. To prove these results, we use stochastic approximation methods, and in particular the ODE method. One difficulty comes from the fact that this method relies on understanding the behaviour at large times of the solution of a non-linear, multi-dimensional ODE.

Research on the feedwater control of a multimodule once through steam generator parallel system

• A complete multimodule parallel OTSG system model is established. • A coordinated control scheme for parallel feedwater of multimodule OTSG is proposed. • Feedforward can improve the response of system in flowrate disturbance. • Wrong control logic may have good control performance, but it hides great danger. Nuclear power plants mostly use the multimodule once-through steam generator (OTSG) parallel system as heat exchange equipment, but the current research mostly focuses on a single OTSG. The huge system brings great challenges to modeling and control system design. In this paper, the ODE method is used to model the main equipment of a multimodule parallel system based on mechanism. The results show that the maximum error is 3.77%. Based on the characteristics of parallel system, a new control strategy is proposed and compared with other control strategies by introducing four typical disturbances. The results show that the proposed feedwater control scheme can effectively solve the coordination between loop valve and module valve of multimodule parallel OTSG system, and the coordination between main feedwater pump and loop valve. Compared with Method 1, the response time of Method 2 in sodium temperature disturbance is shortened by 319 s and the temperature fluctuation is reduced by 1.67 times, but Method 2 has potential safety hazards. Method 1 can improve the system response speed when the sodium flowrate and water flowrate change. Compared with Method 3, the outlet sodium temperature fluctuation of Method 1 is reduced by 10 times and the adjustment time of Method 1 shortens 141.9 s in the case of feedwater flow disturbance.

New multiple-different impressive perceptions for the solitary solution to the magneto-optic waveguides with anti-cubic nonlinearity

In this study, we will establish multiple-impressive different perceptions for the solitary wave solution to the magneto-optic waveguides with anti-cubic nonlinearity (MOWWACN) in the framework of three distinct manners. The three manners are implemented in the same vein and parallel to extract multiple new perceptions for the solitary wave solutions to this model. The first one is the extended simple equation method (ESEM), while the second one is the Paul-Painlevé approach method (PPAM) and the third one is the Riccati-Bernolli Sub ODE method (RBSODM). A good comparison has been constructed not only between our extracted solitary wave solutions which are achieved by these three methods but also with that extracted previously by other authors.

New visions of the soliton solutions to the modified nonlinear Schrodinger equation

In this article, we will study the modified nonlinear Schrödinger equation which represents the propagation of rogue waves in ocean engineering that occur in deep water, usually far out at sea, and are a threat even to capital ships and ocean liners. In related subject for all like waves which occurred at different branches of science which have the same behaviors such as fluid dynamics waveguides that have unexpected large displacements, long-wave limit of a breather (a pulsing mode), telecommunications engineering devices, etc. We will implement new multiple visions for the soliton solution to the modified nonlinear Schrödinger equation (MNLSE) that describes successfully these phenomenon. Studying this important model and its own behavior of the resulting solitons description will contribute effectively in marine safety and improvement the pulse devices which cooperate to protect human life effectively and technology for high-capacity optical communications through which the modern asocial media was built…etc. Three different techniques are invited for this purpose. The first technique is the Paul-Painlevé approach method (PPAM), while the second technique is the Ricatti-Bernolli Sub ODE method (RBSOM). In related subject, the variational iteration method VIM is implemented in the same vein and parallel to construct the numerical solutions corresponding to the achieved soliton solutions via each one of these two methods individually. A good comparison not only between these three visions of soliton solutions but also with that achieved previously by other authors has been demonstrated.

Attractor as a convex combination of a set of attractors

This paper presents an effective approach to constructing numerical attractors of a general class of continuous homogenous dynamical systems: decomposing an attractor as a convex combination of a set of other existing attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical attractor via a convex combination of some existing attractors. Several examples are presented to show the effectiveness of the proposed method.

Complete constant mean curvature hypersurfaces in Euclidean space of dimension four or higher

In this article we provide a general construction when $n\ge3$ for immersed in Euclidean $(n+1)$-space, complete, smooth, constant mean curvature hypersurfaces of finite topological type (in short CMC $n$-hypersurfaces). More precisely our construction converts certain graphs in Euclidean $(n+1)$-space to CMC $n$-hypersurfaces with asymptotically Delaunay ends in two steps: First appropriate small perturbations of the given graph have their vertices replaced by round spherical regions and their edges and rays by Delaunay pieces so that a family of initial smooth hypersurfaces is constructed. One of the initial hypersurfaces is then perturbed to produce the desired CMC $n$-hypersurface which depends on the given family of perturbations of the graph and a small in absolute value parameter $\underline\tau$. This construction is very general because of the abundance of graphs which satisfy the required conditions and because it does not rely on symmetry requirements. For any given $k\ge2$ and $n\ge3$ it allows us to realize infinitely many topological types as CMC $n$-hypersurfaces in $\mathbb R^{n+1}$ with $k$ ends. Moreover for each case there is a plethora of examples reflecting the abundance of the available graphs. This is in sharp contrast with the known examples which in the best of our knowledge are all (generalized) cylindrical obtained by ODE methods and are compact or with two ends. Furthermore we construct embedded examples when $k\ge3$ where the number of possible topological types for each $k$ is finite but tends to $\infty$ as $k\to\infty$.

Complete monotonicity-preserving numerical methods for time fractional ODEs

The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We therefore introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes are at least $A(\pi/2)$ stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs, both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the $\mathcal{L}$1 scheme is $\mathcal{CM}$-preserving, so that the $\mathcal{L}$1 scheme is at least $A(\pi/2)$ stable, which is an improvement on stability analysis for $\mathcal{L}$1 scheme given in Jin, Lazarov and Zhou (IMA J. Numer. Analy. 36:197-221, 2016). The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs are extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.