Initial Value Problem Research Articles

On the adaptive solution of phase‐field problems with A‐stable explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods

AbstractRunge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems with diagonally implicit Runge‐Kutta (DIRK) methods as the most common approach. In this contribution, the new low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher‐order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non‐linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaux to phase‐field problems, solved with the finite‐element method, leading to substantial numerical investigations with an efficient approach for DIRK schemes. The most important aspects are the investigations of stability properties which lead to the novel class of A‐stable ELDIRK methods. In accordance, the study of the convergence orders is presented. A local error estimator is presented, such that adaptive step size control for the new low‐order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two‐dimensional phase‐field simulations based on a Kobayashi‐Warren‐Carter model. The higher‐order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi‐explicit addition to the family of Runge–Kutta time integration schemes.

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.

Existence and uniqueness of solutions of the Koopman–von Neumann equation on bounded domains

Abstract The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman–von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman–von Neumann framework to transport equations is highlighted.

Minimal wave speed and spreading speed in predator-prey systems with stage structure

This article studies the minimal wave speed of traveling wave solutions as well as spreading speed in predator-prey systems. The model divides the prey into the immaturity and maturity, and the predator feeds on the immature individuals. For the traveling wave solutions connecting the predator-free equilibrium with the coexistence state, we consider two different diffusion modes. For both modes, the minimal wave speed is shown by presenting the existence or nonexistence of nontrivial traveling wave solutions for all wave parameters. When the diffusion is local, the minimal wave speed is the spreading speed of the corresponding initial value problem, in which the initial condition implies that the predator is the invader while the prey is the aborigine in the whole habitat. Under the persistence assumption in the corresponding kinetic systems, our conclusion coincides with the fact that the diffusion of the prey may have less effect on the invasion ability of the predator.

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

On One Point Singular Nonlinear Initial Boundary Value Problem for a Fractional Integro-Differential Equation via Fixed Point Theory

In this article, we focus on examining the existence, uniqueness, and continuous dependence of solutions on initial data for a specific initial boundary value problem which mainly arises from one-dimensional quasi-static contact problems in nonlinear thermo-elasticity. This problem concerns a fractional nonlinear singular integro-differential equation of order θ∈[0,1]. The primary methodology involves the application of a fixed point theorem coupled with certain a priori bounds. The feasibility of solving this problem is established under the context of data related to a weighted Sobolev space. Furthermore, an additional result related to the regularity of the solution for the formulated problem is also presented.

Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique α value; instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known.

On explicit ninth‐order, two‐step methods addressing y″=f(x,y)

We present a new family of ninth‐order hybrid explicit Numerov‐type methods, effectively utilizing only eight stages, for solving the special second‐order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well‐known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm.

A NEW ALGORITHM FOR GENERATING POWER SERIES SOLUTIONS FOR A BROAD CLASS OF FRACTIONAL PDES: APPLICATIONS TO INTERESTING PROBLEMS

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

A Non-Standard Finite Difference Schemes for the Solution of Stiff Initial Value Problems

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

Variations of heat equation on the half‐line via the Fokas method

In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat‐mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well‐known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second‐order fluid equation), (ii) a fourth‐order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double‐diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter‐plane with arbitrary, fully non‐homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed‐form solutions will be demonstrated by studying their long‐time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.

A Riemann–Hilbert approach to the existence results for the Benjamin–Ono equation on a half-line

The main problem addressed in this paper is to study the local existence in time of solutions to the non-homogeneous Neumann initial boundary value problem for the Benjamin–Ono equation on a half-line. In this result, we observe the influence of the boundary data on the behavior of solutions. In order to obtain the characterization of the solution it is essential to use the theory concerning the Riemann–Hilbert problem. We prove local existence in time of the solutions.

Mathematical analysis of the motion of a piston in a fluid with density dependent viscosity

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons.

Inverse scattering transform for the higher-order integrable discrete nonlinear Schrödinger equation with nonzero boundary conditions

The inverse scattering transform is a tool for solving the initial value problem of nonlinear equations, and the solution of the initial-value problem by inverse scattering transform proceeds in three steps: direct scattering, time evolution and inverse problem. In this paper, we discuss the higher-order integrable discrete nonlinear Schrödinger equation which is subjected to inverse scattering transform under nonzero boundary conditions, and the corresponding soliton solutions are obtained and illustrated.

Hybrid block formulae whose eigenvalues of Jacobian matrices are close to the imaginary axis of the complex plane

Methods for integrating stiff initial value problems are required to be A-stable. Of great interest are A-stable methods whose Jacobian matrices have their eigenvalues close to the imaginary axis of the complex plane. This class of A-stable methods are very rare. This paper is on the development of a new family of A-stable hybrid block method whose Jacobian matrices possess eigenvalues on the imaginary of the complex plane via interpolation and collocation techniques. The family of methods developed herein are A-stable for order p ≤ 18. Numerical solutions generated by the new method are compared with existing methods in the literature. The numerical results show that the new class of methods are more efficient and accurate.

Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems

Eigenvalue analysis and applications of the Legendre dual-Petrov-Galerkin methods for initial value problems

Existence and Uniqueness of Solution Represented as Fractional Power Series for the Fractional Advection–Dispersion Equation

The fractional advection–dispersion equation is used in groundwater hydrology for modeling the movements of contaminants/solute particles along with flowing groundwater at the seepage velocity in porous media. This model is used for the prediction of the transport of nonreactive dissolved contaminants in groundwater. This paper establishes the existence and the uniqueness of solutions represented as fractional bi-variate power series of some initial-value problems and boundary-value problems for the fractional advection–dispersion equation. Moreover, a method to approximate the solutions using fractional polynomials in two variables and to evaluate the errors in a suitable rectangle is designed. Illustrative examples showing the applicability of the theoretical results are presented.

Large-time behavior of the solutions for an inflow problem of the full compressible Navier–Stokes–Korteweg equations

In this article, we are concerned with the large-time behavior of solutions to an inflow problem for the one-dimensional full compressible Navier–Stokes–Korteweg equations, which can be used to model the compressible viscous fluids with internal capillarity and heat conductivity. Under some suitable assumptions of the far fields and boundary values of the density, the velocity and the absolute temperature, the large-time behavior of the solutions for the initial boundary value problem is govern by the stationary solution and basic waves of the corresponding hyperbolic system. Then the asymptotic stability of the boundary layer, and the superposition of the boundary layer and the three-rarefaction wave are shown provided that the initial perturbation and the strength of the rarefaction wave and stationary wave are small. The proof is mainly based on L2-energy method, some time-decay estimates in Lp-norm for the smoothed rarefaction wave and the space variable decay estimates of the boundary layer. This can be viewed as the first result about the nonlinear stability of the combination of two different wave patterns for the inflow problem of the full compressible Navier–Stokes–Korteweg equations.

Dynamical response of a floating ice sheet due to a forced oscillation in a running stream

This paper explores the response of a floating ice sheet to a forced, time-harmonic oscillatory pressure applied to the ice-covered surface of a uniform, finite depth fluid. The floating ice sheet is modeled as a thin elastic plate following the Euler–Bernoulli beam equation, with the additional consideration of in-plane compressive forces acting on the ice. Employing linear wave theory, the problem is presented as an initial boundary value problem and is solved using Laplace and Fourier transforms to obtain a mathematical expression for the ice sheet's deflection in terms of infinite integrals. These integrals are thereafter solved asymptotically for large time and distance using the stationary phase method. The asymptotic analysis indicates a growing response over time when the poles and stationary points of the phase functions coalesce. The deflection of the floating ice sheet is graphically presented, highlighting the effects of various non-dimensional parameters, such as flexural rigidity, compressive force, uniform current speed, and the angular frequency of the oscillatory pressure. Additionally, the group velocity and phase velocity of flexural gravity waves are derived from the dispersion relation and elucidated using diagrams. The results show that compressive force and current speed significantly influence wave amplitude, enhancing the oscillatory nature, while the flexural rigidity of the elastic plate and the angular frequency of the applied pressure have a substantial impact on the plate's deflection.

Six Points Implicit Block Method with Extra Derivative for Solving Second Order Ordinary Differential Equations

The construction of the six-point block methods with extra derivatives for solving directly is presented in this paper. The suggested block methods concurrently approximate the problem's solution at six points and are formulated using the Hermite Interpolating Polynomial. The block methods do not reduce the equation to a first-order system of initial value problems (IVPs); instead, they directly obtain the numerical solutions. Additionally, the order and zero-stability of the suggested techniques are examined. We present numerical results and draw comparisons with other block methods that are currently in use. The results demonstrate how effective the suggested approaches are at solving second-order IVPs in general.