Solutions Of Ordinary Differential Equations Research Articles

Insight into the study of some nonlinear evolution problems: Applications based on Variation Iteration Method with Laplace

In this study, we look at the solutions of nonlinear partial differential equations and ordinary differential equations. Scientists and engineers have had a hard time coming up with a way to solve nonlinear differential equations. Almost all of the nature’s puzzles have equations that aren’t linear. There aren’t any well-known ways to solve nonlinear equations, and people have tried to improve methods for a certain type of problems. This doesn’t mean, however, that all nonlinear equations can be solved. With this in mind, we’ll look at how well the variation approach works for solving nonlinear DEs. Different problems can be solved well by using different methods. We agree that a nonlinear problem might have more than one answer. Factorization, homotropy analysis, homotropy perturbation, tangent hyperbolic function and trial function are all examples of ways to do this. On the other hand, some of these strategies don’t cover all of the nonlinear problem-solving methods. In this paper, a new method called the variation iterative method with Laplace transformation is used to find a solution to the highly nonlinear evolution of a simple pendulum whose rotation revolves around its fixed position. When the Laplace operator is used to change the Maximum Minimum Approach, Amplitude Frequency Formulation and Variation Iteration Method (VIM) nonlinear oscillators, the results of the analysis are all the same. The method for solving nonlinear oscillators, as well as their time and boundary conditions, can be shown to be correct by comparing analytical results of VIM obtained through the Laplace transformation.

Liquidity Provision with Adverse Selection and Inventory Costs

We study one-shot Nash competition between an arbitrary number of identical dealers that compete for the order flow of a client. The client trades either because of proprietary information, exposure to idiosyncratic risk, or a mix of both trading motives. When quoting their price schedules, the dealers do not know the client’s type but only its distribution, and in turn choose their price quotes to mitigate between adverse selection and inventory costs. Under essentially minimal conditions, we show that a unique symmetric Nash equilibrium exists and can be characterized by the solution of a nonlinear ordinary differential equation.

GenASiS Mathematics: Object-oriented manifolds, operations, and solvers for large-scale physics simulations (version 2)

GenASiSMathematics provides modern Fortran classes furnishing extensible object-oriented functionality for the solution of fields governed by selected partial differential equations. The initial release included extensible object-oriented implementations of simple meshes and the evolution of generic conserved currents thereon. This revision—Version 2 of Mathematics—includes significant reorganization and streamlining of these classes, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs. New version program summaryProgram Title:HomogeneousSpheroid, RectangularContraction, RectangularExpansion, SphericalContraction, SphericalExpansion (example problems illustrating GenASiSMathematics)CPC Library link to program files:https://doi.org/10.17632/mzvxngwtw6.2Developer's repository link:https://github.com/GenASiSCode Ocean capsule:https://codeocean.com/capsule/4106782Licensing provisions: GPLv3Programming language: Modern Fortran; OpenMP (tested with recent versions of GNU Compiler Collection (GCC), Cray Compiler Environment (CCE), IBM XL Fortran compiler)Journal reference of previous version: Computer Physics Communications 222 (2018) 384Does the new version supersede the previous version?: YesReasons for the new version: This version includes significant reorganization and streamlining, higher-order reconstruction by a different method, a Poisson solver, coarsening to avoid Courant time step limitations near coordinate singularities, and the offloading of computational kernels to GPUs.Summary of revisions: The left part of Fig. 1 shows the revised structure of GenASiSMathematics. The Algebra and Calculus divisions contain relatively simple stand-alone functionality for such tasks as root finding, interpolation, numerical integration, and the solution of ordinary differential equations. The major functionality currently provided in Mathematics, which focuses on the solution of selected classes of partial differential equations, is now collected in the CauchyProblems division.The structure of CauchyProblems is displayed in the right part of Fig. 1. The solution of partial differential equations presumes the existence of Manifolds (or ‘spaces’) and Fields thereon whose configuration we seek. The two basic types of Cauchy problems are boundary value problems, or Constraints; and initial value problems, or Evolutions.The structures of Manifolds, Fields, Constraints, and Evolutions are displayed in Fig. 2. Individual coordinate patches or Charts are combined into Atlases in order to represent Manifolds. As for Fields, the most basic functionality for a set of fields on a manifold (including I/O) is representedby classes in FieldSets. Specialized sets of fields needed for Cauchy problems are Geometries, including coarsening functionality to avoid Courant time step limitations near coordinate singularities; and CurrentSets, which represent ‘conserved’—or more properly, when source terms are present, ‘balanced’—currents, that is, densities and their corresponding fluxes. CalculusFields provides integrals of fields on manifolds, including volume and surface integrals used to tally conserved and balanced quantities, as well as spherical and azimuthal averaging. New to this version of Mathematics is the solution of PoissonEquations via multipole expansion, the first capability for Constraints or boundary value problems.The solution of balance equations is the type of initial value problem currently handled by Evolutions. As in Version 1, finite-volume discretization results in a large set of ordinary differential equations solved by the Runge-Kutta method. Now however, the right-hand sides, or Slopes, are handled by variations of a recursive class that allows the construction of hierarchies of terms. Thus complicated equations can be computed and included in I/O in pieces representing various terms, with results assembled in an extensible and quasi-automated fashion. For example, the partial derivatives, and geometric source terms from connection coefficients, can be computed and visualized separately, while being automatically combined to give the full divergence of an energy-momentum tensor. Third-order reconstruction (with limiting) to discretize cell interfaces is now included. Multi-stage Runge-Kutta Steps are then taken by Integrators in order to advance the solution of fields from their initial condition.Relative to Version 1 of Mathematics, the Manifolds and Fields classes are significantly reorganized and streamlined. Previously, classes for fields, geometry, and I/O were interleaved with those of charts and atlases. This provided for chart and atlas classes with self-contained geometry and I/O functionality, but in a manner that proved unwieldy and indeed unnecessary. This has now been disentangled: classes for fields, geometry and I/O are built entirely on top of full manifolds (i.e. classes for atlases).All computational kernels (except those specifically needed for checkpointing, that is, I/O) are coded for offloading to GPUs via OpenMP.Nature of problem: By way of illustrating GenASiSMathematics functionality, solve an example potential problem and imposed advection problems.Solution method: Multipole expansion for solution of the Poisson equation; finite-volume discretization; second- and third-order reconstruction with limiting; HLL Riemann Solver; Runge-Kutta integration.Additional comments including restrictions and unusual features: Uses the MPI [1] and Silo [2] libraries. The example problems named above are not ends in themselves, but serve to illustrate the functionality available though GenASiSMathematics. In addition to these more substantial examples, we provide individual unit test programs for the classes comprised by GenASiSMathematics.GenASiSMathematics is available in the CPC Program Library and also at https://github.com/GenASiS.

Set-Based B-Series

B-series were defined to unify the formalism of solutions for ordinary differential equations defined by series. Runge–Kutta schemes can be seen as truncated B-series, similar to Taylor series. In the prolific domain of reachability analysis, i.e., the process of computing the set of reachable states for a system, many techniques have been proposed without obvious links. In the particular case of uncertain initial conditions and/or parameters in the definition of differential equations, set-based approaches are a natural and elegant method to compute reachable sets. In this paper, an extension to B-series is proposed to merge these techniques in a common formalism—named set-based B-series. We show that the main properties of B-series are preserved. A validated technique, based on Runge–Kutta methods, able to compute such series, is presented. Experiments are provided in order to illustrate the proposed approach.

Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method

In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.

The role of problem context familiarity in modelling first-order ordinary differential equations

Based on the unpredictable effect of context familiarity when students solve real-world problems, this work aims to analyse how certain contexts can be used by pre-service mathematics teachers in the representation and real-world verification of a first-order mathematical model in the classroom in the subject of Ordinary Differential Equations. Specifically, this paper reports a classroom experience in which pre-service mathematics teachers compared the solution of a first-order ordinary differential equations (ODE) with a real-world experimental model. Using documentary records (i.e., students´ hand-written solutions and field notes) and a questionnaire on students´ perceptions on this classroom experience, qualitative results indicated that the pre-service mathematics teachers’ familiarity with an authentic context was a fundamental factor they chose a real-world model to represent the solution of a first-order ODE. Our analysis of the results highlights the importance of integrating familiar real-world contexts for pre-service mathematics teachers to model a first-order ODE, which is one of the fundamental principles of STEM disciplines.

Multivalue second derivative collocation methods

We introduce multivalue second derivative collocation methods for the numerical solution of stiff ordinary differential equations, also arising from the spatial discretization of time dependent partial differential equations. The uniform order of convergence of the methods is discussed and continuous order conditions are derived. We construct methods of orders up to eight with desirable stability properties. Numerical experiments are given, validating the theoretical results and illustrating the efficiency and capability of the proposed methods in solving stiff problems without any reduction of the order of convergence, unlike A–stable Runge–Kutta methods.

Semi-numerical treatment of MHD Maxwell nanofluid rotating flow on a stretching sheet with the presence of mass transpiration, heat source/sink, and chemical reaction

ABSTRACT The study of the flow of heat and mass transfer in the presence of suction and injection has many scientific and engineering applications such as in aerodynamics and space sciences. A significant role of suction/injection is observed in controlling fluid flow along a boundary layer surface and it is also affecting the heat transfer rate in the mass transfer cooling process. In the present study, we investigate suction/injection effects and heat transfer features of Maxwell nanofluid along a stretched sheet in addition to a chemical reaction, magnetohydrodynamics, and heat source/sink. The partial differential equations describing the physical phenomena of the problem under consideration are converted into non-dimensional ordinary differential equations (ODEs) through an appropriate set of similarity transformations. Transformed ODEs solution is obtained with a semi-optimal homotopy analysis method (OHAM). The influencing behavior of suction and injection, as well as of other parameters, on the velocity, temperature, and concentration fields are shown through graphs and tables. The horizontal velocity and temperature field show a decreasing behavior for both suction and injection while opposite trends are noticed in transverse velocity and concentration profile.

Convective transport via thermophoresis and brownian forces in MHD Hiemenz stagnation-point flow

The heat and mass transfer in normal impingement of the Hiemenz stagnation-point flow of a viscous nanofluid on a bidirectional stretching sheet under the influence of the magnetic field is investigated. In a recent study, Weidman (Meccanica, vol. 53, p.833, 2018) has analyzed the phenomenon of the Hiemenz stagnation-point flow over the biaxially stretching sheet. Dual solutions have been found in this study for the various values of [Formula: see text] (streamwise stretching rate) and [Formula: see text] (cross-stream stretching rate). In the present work, the mechanism of thermal and solutal energy transport in the Hiemenz stagnation-point flow of nanoliquid is studied in the view of thermophoretic and Brownian forces. The solutions of similar ordinary differential equations (ODEs) for flow and energy transport in viscous nanofluid are computed with help of bvp4c numerical technique. The impact of dimensionless parameters on flow field and energy transport is presented graphically and discussed with physical reasoning. The outcomes reveal that the higher magnitude of Lorentz force declines the flow field in cross-stream and vertical directions, but conflicting behavior is noted for streamwise velocity component in the case of [Formula: see text] and [Formula: see text]. Both thermophoresis and Brownian motion of nanoparticles enhance the energy transport in Hiemenz flow of viscous fluid. The results of the present study are validated through the shear stress parameters with a previously published study.

Inverse Flood Routing Using Simplified Flow Equations

The paper considers the problem of inverse flood routing in reservoir operation strategy. The aim of the work is to investigate the possibility of determining the hydrograph at the upstream end based on the hydrograph required at the downstream end using simplified open channel flow models. To accomplish this, the linear kinematic wave equation, the diffusive wave equation and the linear Muskingum equation are considered. To achieve the hydrograph at the upstream end, an inverse solution of the afore mentioned equations with backward integration in the x direction is carried out. The numerical solution of the kinematic wave equation and the Muskingum equation bases on the finite difference scheme. It is shown that both these equations are able to provide satisfying results because of their exceptional properties related to numerical diffusion. In the paper, an alternative approach to solve the inverse routing using the diffusive wave model is also presented. To this end, it is described by a convolution which involves the instantaneous unit hydrograph (IUH) corresponding to the linear diffusive wave equation. Consequently, instead of a solution of partial or ordinary differential equations, the integral equation with Laguerre polynomials, used for the expansion of the upstream hydrograph, is solved. It was shown that the convolution approach is more reliable comparing to the inverse solution of the simplified models in the form of differential equations.

A modified Lyapunov method and its applications to ODE

Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard‐Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations.

Exact Solutions and Non-Traveling Wave Solutions of the (2+1)-Dimensional Boussinesq Equation

By the extended (G′G) method and the improved tanh function method, the exact solutions of the (2+1) dimensional Boussinesq equation are studied. Firstly, with the help of the solutions of the nonlinear ordinary differential equation, we obtain the new traveling wave exact solutions of the equation by the homogeneous equilibrium principle and the extended (G′G) method. Secondly, by constructing the new ansatz solutions and applying the improved tanh function method, many non-traveling wave exact solutions of the equation are given. The solutions mainly include hyperbolic, trigonometric and rational functions, which reflect different types of solutions for nonlinear waves. Finally, we discuss the effects of these solutions on the formation of rogue waves according to the numerical simulation.

Monic Chebyshev pseudospectral differentiation matrices for higher-order IVPs and BVPs: applications to certain types of real-life problems

We introduce new differentiation matrices based on the pseudospectral collocation method. Monic Chebyshev polynomials (MCPs) were used as trial functions in differentiation matrices (D-matrices). Those matrices have been used to approximate the solutions of higher-order ordinary differential equations (H-ODEs). Two techniques will be used in this work. The first technique is a direct approximation of the H-ODE. While the second technique depends on transforming the H-ODE into a system of lower order ODEs. We discuss the error analysis of these D-matrices in-depth. Also, the approximation and truncation error convergence have been presented to improve the error analysis. Some numerical test functions and examples are illustrated to show the constructed D-matrices’ efficiency and accuracy.

Meromorphic solutions of autonomous ordinary differential equations without the finiteness property

We consider the problem of finding meromorphic solutions of autonomous algebraic ordinary differential equations possessing an infinite number of local solutions given by Laurent series in neighborhoods of poles. We present an algebraic-geometric method for finding all meromorphic solutions that are elliptic or rational in the exponential function. As an example, we classify such families of meromorphic solutions for fourth-order ordinary differential equations arising via integrating once traveling-wave reductions of the generalized Rosenau–Korteweg–de Vries equation and the fifth-order Korteweg–de Vries–Burgers equation. Solutions important from a physical point of view are presented.

Stochastic Parareal: An Application of Probabilistic Methods to Time-Parallelization

Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a solution in a fixed number of iterations using a predictor-corrector, stopping once a tolerance is met. This iterative process combines solutions located by inexpensive (coarse resolution) and expensive (fine resolution) numerical integrators. In this paper, we introduce a stochastic parareal algorithm aimed at accelerating the convergence of the deterministic parareal algorithm. Instead of providing the predictor-corrector with a deterministically located set of initial values, the stochastic algorithm samples initial values from dynamically varying probability distributions in each temporal sub-interval. All samples are then propagated in parallel using the expensive integrator. The set of sampled initial values yielding the most continuous (smoothest) trajectory across consecutive sub-intervals are fed into the predictor-corrector, converging in fewer iterations than the deterministic algorithm with a given probability. The performance of the stochastic algorithm, implemented using various probability distributions, is illustrated on low-dimensional systems of ordinary differential equations (ODEs). We provide numerical evidence that when the number of sampled initial values is large enough, stochastic parareal converges almost certainly in fewer iterations than the deterministic algorithm, maintaining solution accuracy. Given its stochastic nature, we also highlight that multiple simulations of stochastic parareal return a distribution of solutions that can represent a measure of uncertainty over the ODE solution.

A family of r-points 1-block implicit methods with optimized region of stability for stiff initial value problems in ordinary differential equations

In this paper, we proposed a family of r-points 1-block implicit methods with optimized region of stability. This family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques. The block methods are consistence, zero stable, and A-stable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs. Numerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations. The methods have been found to show competitiveness with other numerical methods.

On zeros of solutions of ordinary second order differential equations

On zeros of solutions of ordinary second order differential equations

Discrete Dynamical System Approaches for Boolean Polynomial Optimization

In this article, we discuss the numerical solution of Boolean polynomial programs by algorithms borrowing from numerical methods for differential equations, namely the Houbolt scheme, the Lie scheme, and a Runge-Kutta scheme. We first introduce a quartic penalty functional (of Ginzburg-Landau type) to approximate the Boolean program by a continuous one and prove some convergence results as the penalty parameter $$\varepsilon $$ converges to 0. We prove also that, under reasonable assumptions, the distance between local minimizers of the penalized problem and the set $$\{\pm 1\}^n$$ is of order $$O(\sqrt{n}\varepsilon )$$ . Next, we introduce algorithms for the numerical solution of the penalized problem, these algorithms relying on the Houbolt, Lie and Runge-Kutta schemes, classical methods for the numerical solution of ordinary or partial differential equations. We performed numerical experiments to investigate the impact of various parameters on the convergence of the algorithms. We have tested our ODE approaches and compared with the classical nonlinear optimization solver IPOPT and a quadratic binary formulation approach (QB-G) as well as an exhaustive method using parallel computing techniques. The numerical results on various datasets (including small and large-scale randomly generated synthetic datasets of general Boolean polynomial optimization problems, and a large-scale heterogeneous MQLib benchmark dataset of Max-Cut and Quadratic Unconstrained Binary Optimization (QUBO) problems) show good performances for our ODE approaches. As a result, our ODE algorithms often converge faster than the other compared methods to better integer solutions of the Boolean program.

Resource-Efficient Use of Modern Processor Architectures For Numerically Solving Cardiac Ionic Cell Models.

A central component in simulating cardiac electrophysiology is the numerical solution of nonlinear ordinary differential equations, also called cardiac ionic cell models, that describe cross-cell-membrane ion transport. Biophysically detailed cell models often require a considerable amount of computation, including calls to special mathematical functions. This paper systematically studies how to efficiently use modern multicore CPUs for this costly computational task. We start by investigating the code restructurings needed to effectively enable compiler-supported SIMD vectorisation, which is the most important performance booster in this context. It is found that suitable OpenMP directives are sufficient for achieving both vectorisation and parallelisation. We then continue with an evaluation of the performance optimisation technique of using lookup tables. Due to increased challenges for automated vectorisation, the obtainable benefits of lookup tables are dependent on the hardware platforms chosen. Throughout the study, we report detailed time measurements obtained on Intel Xeon, Xeon Phi, AMD Epyc and two ARM processors including Fujitsu A64FX, while attention is also paid to the impact of SIMD vectorisation and lookup tables on the computational accuracy. As a realistic example, the benefits of performance enhancement are demonstrated by a 109-run ensemble on the Oakforest-PACS system, where code restructurings and SIMD vectorisation yield an 84% reduction in computing time, corresponding to 63,270 node hours.

Discounted optimal stopping problems in first-passage time models with random thresholds

AbstractWe derive closed-form solutions to some discounted optimal stopping problems related to the perpetual American cancellable dividend-paying put and call option pricing problems in an extension of the Black–Merton–Scholes model. The cancellation times are assumed to occur when the underlying risky asset price process hits some unobservable random thresholds. The optimal stopping times are shown to be the first times at which the asset price reaches stochastic boundaries depending on the current values of its running maximum and minimum processes. The proof is based on the reduction of the original optimal stopping problems to the associated free-boundary problems and the solution of the latter problems by means of the smooth-fit and modified normal-reflection conditions. We show that the optimal stopping boundaries are characterised as the maximal and minimal solutions of certain first-order nonlinear ordinary differential equations.