Evolution Partial Differential Equations Research Articles

Compactons in Higher-Order Nesterenko's-Type Equations

Abstract A model describing propagation of waves in a prestressed granular media is considered. The model, having the form of evolutionary partial differential equation (PDE), is obtained from the system of ordinary differential equations (ODEs) describing dynamics of a chain of prestressed granules by means of formal asymptotic expansion. It is shown in our previous papers that in the lowest asymptotic approximation, in which both nonlinear effects and the presence of media structure are taken into account, the model equation possesses traveling wave (TW) solutions with compact support (compactons) manifesting soliton properties. In this paper, we study a higher-order evolutionary PDE obtained by taking into account previously discarded terms of the asymptotic expansion, as well as another PDE (called analogue), differing from the original one in the values of parameters and having compacton solutions expressed in analytical form. Numerical and analytical studies of both the higher-order model and its analogue allow to conclude that both models have compacton solutions exhibiting some properties of “true” solitons. This, in turn, testifies the stability of the previously used model with respect to the inclusion of the discarded terms of the asymptotic expansion.

Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review

Hyperbolic systems with stiff relaxation constitute a wide class of evolutionary partial differential equations which describe several physical phenomena, ranging from gas dynamics to kinetic theory, from semiconductor modeling to traffic flow. Peculiar features of such systems is the presence of a small parameter that determines the smallest time scale of the system. As such parameter vanishes, the system relaxes to a different one with a smaller number of equations, and sometime of a different mathematical nature. The numerical solution of such systems may present some challenges, in particular if one is interested in capturing all regimes with the same numerical method, including the one in which the small parameter vanishes (relaxed system). The design, analysis and application of numerical schemes which are robust enough to solve this class of systems for arbitrary value of the small parameter is the subject of the current paper. We start presenting different classes of hyperbolic systems with relaxation, illustrate the properties of implicit–explicit (IMEX) Runge–Kutta schemes which are adopted for the construction of efficient methods for the numerical solution of the systems, and then illustrate how to apply IMEX schemes for the construction of asymptotic preserving schemes, i.e. scheme which correctly capture the behavior of the systems even when the relaxation parameter vanishes.

Lie symmetry analysis, multiple exp-function method and conservation laws for the (2+1)-dimensional Boussinesq equation

In this study, we take into account the (2 + 1)-dimensional Boussinesq equation, a nonlinear evolution partial differential equation that describes how gravity waves move across the surface of the ocean. The symmetry reductions and group invariant precise solutions are systematically determined using the Lie symmetry analysis. We derive the precise multiple wave solutions using the multiple exp-function method, and then, using the multiplier method, we give the conservation laws. The dynamics of complicated waves and their interplay are faithfully recreated by the findings.

ДЕЯКІ КРАЙОВІ ЗАДАЧІ ДРОБОВО-ДИФЕРЕНЦІЙНОЇ ФІЛЬТРАЦІЙНОЇ ДИНАМІКИ ЩОДО БІПАРАБОЛІЧНОЇ МАТЕМАТИЧНОЇ МОДЕЛІ

Closed-form solutions are obtained to some one-dimensional boundary-value problems for modeling anomalous filtration dynamics in a layered geoporous medium, posed within the framework of the fractional-differential generalization of the biparabolic evolutionary partial differential equation of the fourth order. In particular, the formulation and solution of the direct and inverse model boundary-value problems of geofiltration dynamics based on the mathematical model with conjugation conditions are presented, and the conditions of the existence of regular solutions to these problems are defined. Keywords: mathematical modeling, fractional-differential dynamics of geofiltration processes, nonclassical models, biparabolic evolutionary equation, the fractional-differential analog of the biparabolic evolutionary equation, nonstationary boundary-value problems on a finite interval, direct and inverse problems, conjugation conditions, closed-form solutions.

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

Rimming flow in a rotating horizontal cylinder with phase change at the interface

The two-dimensional solutions and stability analysis are presented for an evaporating thin viscous liquid film flowing inside a uniformly heated rotating horizontal cylinder. A non-linear, fourth-order, partial differential evolution equation is obtained by simplifying mass, momentum, and energy conservation equations within the lubrication approximation. The effect of evaporation, gravity, viscous drag, surface tension, thermocapillary stress, and intermolecular forces has been taken into account. The numerical solutions of the model are validated against the existing experimental as well as the numerical results, along with the analytical result for the limiting cases of the present model. The film thickness model is solved to elucidate two-dimensional spatiotemporal solutions and their stability for a wide range of thermal and other parameters. The evaporative mass flux at the liquid–air interface results in unsteady solutions which are oscillatory in nature, and the amplitude of the oscillations increases with an increase in the evaporative flux. The film ruptures after some time and the rupture time is found to be inversely proportional to the evaporation number, a non-dimensional number quantifying the rate of evaporation. The linear stability analysis shows that the thermocapillary stresses as well as the long-range molecular forces destabilize the film. A negative disjoining pressure is shown to reduce the rupture time and vice versa. Evaporation (condensation) plays a destabilizing (stabilizing) role in the thin film flow. Non-linear computations are carried out for the steady profiles, validating the growth rates obtained from the linear stability analysis.

Generalized conditional symmetries and pre-Hamiltonian operators

In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand–Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs.

Analysis of soliton interactions of modified Korteweg-de Vries equation using conserved quantities

In this paper, the conservative quantities are used to develop an approximate method to calculate the merged waveform shape of the solitary waves described by modified Korteweg–de Vries (mKdV) equation. With this method, we can efficiently and effectively capture the physics of the complicated merging phenomena when two solitary waves described by the nonlinear evolution partial differential equation merge at the maximum without the need to solve the equation in detail. This offers a simple and robust tool to analyse the interactions between solitons and to benchmark the results obtained by the asymptotic and numerical methods. It is expected that the approximate analysis demonstrated in this paper can be applied to a series of nonlinear evolution equations to simulate various solitary wave interaction problems. In future, our goal is to extend this simple method to other nonlinear wave evolution phenomena.

Enthalpy effect on the kinetics of concurrent nucleation and chemical aging of aqueous organic aerosols: The stage of thermal relaxation

The size and composition distribution of an ensemble of aqueous organic droplets, evolving via nucleation and concomitant chemical aging, may be affected by the latent heat of condensation and enthalpy of heterogeneous chemical reactions, so the temperature of the droplet may deviate from the air temperature and thus become an independent variable of its state (additional to its size and composition variables). Using the formalism of the classical nucleation theory, we derive a partial differential equation for the temporal evolution of the distribution of an ensemble of such droplets with respect to all their variables of state via Taylor series expansions of the corresponding multidimensional discrete equation of balance, describing the material and heat exchange between droplets and air. The resulting kinetic equation goes beyond the framework of the Fokker-Planck approximation with respect to the temperature variable. A hierarchy of time scales of nonisothermal nucleation and concomitant chemical aging of aqueous organic aerosols is established and an analytical description of their thermal relaxation stage is developed, allowing one to estimate the characteristic time of the establishment of the equilibrium distribution of aerosol particles with respect to their temperatures. Theoretical results are illustrated with numerical calculations for the concurrent nucleation and chemical aging of model aqueous hydrophilic-hydrophobic organic aerosols in air.

Higher-dimensional integrable deformations of the modified KdV equation

The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability. The well-known modified KdV equation is a prototypical example of an integrable evolution equation in one spatial dimension. Do there exist integrable analogs of the modified KdV equation in higher spatial dimensions? In what follows, we present a positive answer to this question. In particular, rewriting the (1+1)-dimensional integrable modified KdV equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations. Further, we illustrate this idea with examples from the modified KdV hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.

Green’s function for solving initial-boundary value problem of evolutionary partial differential equations

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs’. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green’s functions for the corresponding elliptic PDEs. Finally, we obtain Green’s functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green’s functions of wave equations in [Formula: see text], Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.

Pre-training strategy for solving evolution equations based on physics-informed neural networks

The physics informed neural network (PINN) is a promising method for solving time-evolution partial differential equations (PDEs). However, the standard PINN method may fail to solve the PDEs with strongly nonlinear characteristics or those with high-frequency solutions. To address this problem, we propose a novel method named pre-training PINN (PT-PINN) which can improve the convergence and accuracy of the standard PINN method by combining with the resampling strategy and the existing optimizer combination technique. The PT-PINN method transforms the difficult problem on the entire time domain to relatively simple problems defined on small subdomains. The neural network trained on small subdomains provides the neural network initialization and extra supervised learning data for the problems on larger subdomains or on the entire time-domain. Numerical experiments show that the PT-PINN successfully solves the evolution PDEs with strong non-linearity and/or high frequency solutions, including the strongly nonlinear heat equation, the Allen-Cahn equation, the convection equation with high-frequency solutions and so on. The PT-PINN method is a reliable and competitive method for solving the time-evolution PDEs.

High-Order Semi-implicit Schemes for Evolutionary Partial Differential Equations with Higher Order Derivatives

The aim of this work is to apply a semi-implicit (SI) strategy in an implicit-explicit (IMEX) Runge–Kutta (RK) setting introduced in Boscarino et al. (J Sci Comput 68:975–1001, 2016) to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy gives a great flexibility to treat these equations, and allows the construction of simple linearly implicit schemes without any Newton’s iteration. Furthermore, the SI IMEX-RK schemes so designed does not need any severe time step restriction that usually one has using explicit methods for the stability, i.e. Delta t = {mathcal {O}}(Delta t^k) for the kth (k ge 2) order PDEs. For the space discretization, this strategy is combined with finite difference schemes. We illustrate the effectiveness of the schemes with many applications to dissipative, dispersive and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and can achieve optimal orders of accuracy.

Simulations of the one and two dimensional nonlinear evolutionary partial differential equations: A numerical study

In this work a hybrid scheme is proposed for the numerical study of various evolutionary partial differential equations (EPDEs). In proposed strategy, temporal derivatives are estimated via first and second order finite differences while the solution and spatial derivatives are approximated by Lucas and Fibonacci polynomials. Further, the collocation approach is applied which convert the EPDEs, to the system of coupled linear equations which are simple to solve. For non-linear problems, Quasilinearization is used to tackle nonlinearity. The scheme is implemented to solve different EPDEs which include, one dimensional non-linear Boussinesq, Hunter–Saxton, wave-like and two dimensional linear and non-linear wave like equations. Accuracy of the scheme is portrayed by computing L∞,L2 error norms and the relative error. Moreover, the calculated outcomes are compared with previously existing results in literature. Simulation demonstrates that the scheme works well for the mentioned EPDEs.

On Complete Group Classification of Time Fractional Systems Evolution Differential Equation with a Constant Delay

A fractional order system of evolution partial differential equations with a constant delay is considered. By exploiting the Lie symmetry method, we give a complete group classification of the system. Furthermore, we establish the corresponding symmetry reductions and construct some analytical solutions to the system.

Error estimation for the time to a threshold value in evolutionary partial differential equations

We develop an a posteriori error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals of the solution. Taylor’s theorem and an adjoint-based a posteriori analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Existence of classical solutions for a class of nonlinear impulsive evolution partial differential equations

This paper is devoted to the study of a class of impulsive nonlinear evolution partial differential equations. We give new results about existence and multiplicity of global classical solutions. The method used is based on the use of fixed points for the sum of two operators. Our main results will be illustrated by an application to an impulsive Burgers equation.

Dynamics of Evolutionary Differential Equations with Several Spatial Variables

The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given.

Uniqueness for Riccati equations with application to the optimal boundary control of composite systems of evolutionary partial differential equations

Uniqueness for Riccati equations with application to the optimal boundary control of composite systems of evolutionary partial differential equations

Estimates for evolutionary partial differential equations in classical function spaces

Abstract We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.