Integrable Partial Differential Equations Research Articles

Fixed Point Theorems of Superlinear Operators with Applications

In this paper, by using the partial order method and monotone iterative techniques, the existence and uniqueness of fixed points for a class of superlinear operators are studied, without requiring any compactness or continuity. As corollaries, the new fixed point theorems for α -convex operators α > 1 , e -convex operators, positive α homogeneous operator α > 1 , generalized e -convex operator, and convex operators are obtained. The results are applied to nonlinear integral equations and partial differential equations.

Liouville-type theorems for higher-order Lane–Emden system in exterior domains

In this paper, we are mainly concerned with the following system in an exterior domains: [Formula: see text] where [Formula: see text], [Formula: see text] is an integer, [Formula: see text], and [Formula: see text] is the polyharmonic operator. We prove the nonexistence of positive solutions to the above system for [Formula: see text] if [Formula: see text], and [Formula: see text] if [Formula: see text]. The novelty of the paper is that we do not ask [Formula: see text] satisfy any symmetry and asymptotic conditions at infinity. By proving the superharmonic properties of the solutions, we establish the equivalence between systems of partial differential equations (PDEs) and integral equations (IEs), then the method of scaling sphere in integral form can be applied to prove the nonexistence of the solutions.

From Single to Collective Motion of Social Amoebae: A Computational Study of Interacting Cells

The coupling of the internal mechanisms of cell polarization to cell shape deformations and subsequent cell crawling poses many interdisciplinary scientific challenges. Several mathematical approaches have been proposed to model the coupling of both processes, where one of the most successful methods relies on a phase field that encodes the morphology of the cell, together with the integration of partial differential equations that account for the polarization mechanism inside the cell domain as defined by the phase field. This approach has been previously employed to model the motion of single cells of the social amoeba Dictyostelium discoideum, a widely used model organism to study actin-driven motility and chemotaxis of eukaryotic cells. Besides single cell motility, Dictyostelium discoideum is also well-known for its collective behavior. Here, we extend the previously introduced model for single cell motility to describe the collective motion of large populations of interacting amoebae by including repulsive interactions between the cells. We performed numerical simulations of this model, first characterizing the motion of single cells in terms of their polarity and velocity vectors. We then systematically studied the collisions between two cells that provided the basic interaction scenarios also observed in larger ensembles of interacting amoebae. Finally, the relevance of the cell density was analyzed, revealing a systematic decrease of the motility with density, associated with the formation of transient cell clusters that emerge in this system even though our model does not include any attractive interactions between cells. This model is a prototypical active matter system for the investigation of the emergent collective dynamics of deformable, self-driven cells with a highly complex, nonlinear coupling of cell shape deformations, self-propulsion and repulsive cell-cell interactions. Understanding these self-organization processes of cells like their autonomous aggregation is of high relevance as collective amoeboid motility is part of wound healing, embryonic morphogenesis or pathological processes like the spreading of metastatic cancer cells.

Research on Time Delay Feedback Control of Lateral Vibration Axially Moving Viscoelastic Beam

The nonlinear dynamic behavior of a viscoelastic beam in axially variable motion under time delay is studied. Considering that the axial velocity pulsation and radial tension change periodically with time under the action of time delay, the viscous damping and finite support stiffness are taken into account, and the viscoelastic constitutive relationship adopts Kelvin model. A mathematical model of lateral vibration of an axially moving viscoelastic beam with changes in axial velocity and tension under time delay is established, and a partial differential-integral control equation describing the lateral nonlinear vibration of an axially moving beam is given. Based on the Galerkin truncation method, the control equations established based on the mechanical model are discretized, so that the direct multi-scale method obtains the numerical solution of the lateral nonlinear vibration of the axially moving beam under the action of time delay, and determines the nonlinear dynamic behavior of the system under the action of time delay. By analyzing the numerical solution of the vibration displacement and velocity at the midpoint of the beam, the whole process of the average velocity of the axially moving structure along the axis, the amplitude of the disturbance tension and the change of the viscoelastic coefficient is simulated. It provides a numerical theoretical basis for the study of the nonlinear dynamic behavior of axially moving beams under time delay.Through the research in this article, the introduction of time delay is studied, and the time delay is used to analyze the axial force that changes along the radial direction of the beam caused by the speed pulsation, and the theoretical framework for nonlinear vibration analysis of the axially moving viscoelastic beam is constructed to expand the time delay. The application scope of the nonlinear vibration theory and nonlinear dynamics theory, the development and improvement of the approximate analysis and numerical solution of the nonlinear continuum (especially the gyro body) under the action of time delay, to provide a theoretical basis for the analysis and design of the corresponding engineering system and technical reserves.

Machine Learning for Scientific Imaging 2022 Conference Overview and Papers Program

Machine learning for scientific imaging is a rapidly growing area of research used to characterize physical, material, chemical, and biological processes in both large and small scale scientific experiments. Physics inspired machine learning differs from more general machine learning research in that it emphasizes quantitative reproducibility and the incorporation of physical models. ML methods used for scientific imaging typically incorporate physics-based imaging processes or physics-based models of the underlying data. These models can be based on partial differential equations (PDEs), integral equations, symmetries or other regularity conditions in two or more dimensions. Physics aware models enhance the ability of the ML methods to generalize and robustly operate in the presence of modeling error, incomplete data, and measurement uncertainty. Contributions to the conference are solicited on topics ranging from fundamental theoretical advances to detailed implementations and novel applications for scientific discovery.

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.

Modeling and analysis of structured population in malaria

To obtain clear information about the dynamics of malaria in connection with drug resistance, a quantitative model was developed in the form of a dynamical system for structured populations. Densities for the drug sensitive and resistant parasites were considered as two structured variables. The model system consists of four integral-partial differential equations with implicit boundary conditions. As the main result, the existence and uniqueness of nonnegative solution under a sufficient condition were proven, based on our established framework. The relevance of the model system for better understanding vector-borne diseases with drug resistance was discussed.

Two approximated techniques for solving of system of two-dimensional partial integral differential equations with weakly singular kernels

Two approximated techniques for solving of system of two-dimensional partial integral differential equations with weakly singular kernels

A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a general formula that produces a pre-symplectic operator from a non-gradient adjoint-symmetry. These formulas are illustrated by several examples of linear PDEs and integrable nonlinear PDEs. Additionally, a classification of quasilinear second-order PDEs admitting a multiplicative symmetry recursion operator through the first formula is presented.

Partial differential integral equation model for pricing American option under multi state regime switching with jumps

AbstractIn this paper, we consider a two dimensional partial differential integral equation (PDIE) model for pricing American option. A nonlinear rationality parameter function for two asset problems is introduced to deal with the free boundary. The rationality parameter function is added in the PDIEs used for pricing American option problems under multi‐state regime switching with jumps. The resulting two dimensional nonlinear system of PDIE is then numerically solved. Based on real poles rational approximation, a strongly stable highly efficient and reliable method is developed to solve such complicated systems of PIDEs. The method is build in a predictor corrector style which makes it linearly implicit, therefore, avoids solving nonlinear systems of equations at each time step in all regimes. The method is seen to maintain the stability and convergence for large jump sizes and high volatility in each regime. The impact of regime switching on option prices corresponding to different values interest rate, volatility, and rationality parameter is computed, illustrated by graphs and given in the tables. Convergence results in each regime are presented and time evolution graphs are given to show the effectiveness and reliability of the method.

КВАЗИКЛАССИЧЕСКИЙ ПРЕДЕЛ УРАВНЕНИЙ ШРЕДИНГЕРА-МАКСВЕЛЛА-БЛОХА

The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

Output feedback compensation to state and measurement delays for a first-order hyperbolic PIDE with recycle

This paper develops a novel observer-based output-feedback control law for a class of interconnected first-order hyperbolic partial integral differential equations (PIDEs) with in-domain, non-local coupling terms and measurement delay compensation. A transport PDE is introduced to account for in-domain recycle delay which results in an extended spatial domain with the PIDE and transport PDE being cascaded. Since standard backstepping approach cannot be applied, an affine Volterra integral transform is introduced which allows for the construction of a boundary controller for the PIDE and delay compensation. An observer backstepping transformation containing four kernels is realized, reconstructing the states throughout the delayed measurement. By the use of the backstepping method, a set of coupled kernel equations where one of the kernels has two boundary conditions is established. The existence and invertibility for the control and observer transformations are proved and the finite-time stability for the output system is established while results are illustrated by numerical simulations.

ADI method of credit spread option pricing based on jump-diffusion model

As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral differential equation. Since the integral part might bring some difficulties, we put forward a fairly new numerical scheme based on the alternating direction implicit method. In the remainder of the article, we discuss consistency, stability, and convergence of the proposed approach. As the final step, with the help of the MATLAB program, we provide numerical results of implementing our method on the governing equation.

Stochastic viscosity solutions for stochastic integral-partial differential equations

In order to study stochastic viscosity solutions for a class of semilinear stochastic integral-partial differential equations (SIPDEs), a new class of generalized backward doubly stochastic differential equations with general jumps is investigated. The definition of stochastic viscosity solutions of SIPDEs is introduced. A probabilistic representation for stochastic viscosity solutions of semilinear SIPDEs with nonlinear Neumann boundary conditions is given.

Lump, rogue wave, multi-waves and Homoclinic breather solutions for (2+1)-Modified Veronese Web equation

This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function [Formula: see text] in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

Model based learning for efficient modelling of heat transfer dynamics

Predictive models are crucially relevant for the design and control of manufacturing processes, particularly when process quality is highly sensitive to unknown physical parameters. This paper describes a method to model industrial processes using neural networks and physical prior knowledge. The approach is applied to heat transfer problems, particularly relevant for additive manufacturing processes, and the results are compared to alternative methodologies. Obtained results show that the integration of partial differential equations in the neural network model leads to reduced amounts of required training data and increases the model stability. Such outcomes represent promising characteristics for novel model predictive control strategies.

A Two-Dimensional Map Derived From An Ordinary Difference Equation of mKdV and Its Properties

The discrete modified Korteweg–de Vries (mKdV) is a class of discrete integrable systems that may be distinguished as integrable partial difference equations (PΔE) and integrable ordinary difference equations (OΔE). By considering traveling wave solutions, the OΔE mKdV can be obtained from PΔE mKdV. Meanwhile, a mapping can be constructed from an OΔE mKdV. In this paper, we will focus on producing a new map using a process (replacement), the interchange of a single parameter, and an integral and investigate its properties.

An optimal system of group-invariant solutions and conserved quantities of a nonlinear fifth-order integrable equation

Abstract In this work, we perform Lie group analysis on a fifth-order integrable nonlinear partial differential equation, which was recently introduced in the literature and contains two dispersive terms. We determine a one-parameter group of transformations, an optimal system of group invariant solutions, and derive the corresponding analytic solutions. Topological kink, periodic and power series solutions are obtained. The existence of a variational principle for the underlying equation is proven using Helmholtz conditions and, thereafter, both local and nonlocal conserved quantities are obtained by utilising Noether’s theorem and a homotopy integral approach.

Lerch matrix collocation method for 2D and 3D Volterra type integral and second order partial integro differential equations together with an alternative error analysis and convergence criterion based on residual functions

In this study, second order linear Volterra partial integro-differential equation with two- and three-dimensional are solved by collocation method based on Lerch polynomials. This method is composed of the operational matrix and collocation methods, which are based upon the matrix forms of the Lerch polynomials with the parameter $\lambda$ and Taylor polynomials, and their derivatives and integrals. The approximate solutions of the mentioned equations are investigated in terms of the Lerch polynomials the different values of $\lambda$. Also, to verify the accuracy and efficiency of the present method, an alternative convergence criterion along with error analysis depending on residual function is enhanced. Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.