Partial Integro-differential Equations Research Articles

Variable Piecewise Interpolation Solution of the Transport Equation

In this paper, we construct a piecewise interpolation method of approximate solution of the transport equation based on the Newton interpolation polynomial of two variables. We transform the polynomial to the algebraic form with numerical coefficients; this leads us to a sequence of iterations, which improves the accuracy of the approximation. The method is implemented in software and numerical experiments are performed. The possibility of generalizations to systems of partial differential equations and integro-differential equations is discussed.

Model-Based Feedforward Control of an Intra- and Interspecific Competitive Population System

Intra-and interspecific competitive population systems are relevant for a variety of applications, such as bioreactors or wastewater treatment plants. These systems are described by coupled hyperbolic semilinear integro partial differential equations with non-local integral boundary conditions. This type of population system has not been considered in the context of control theory in the literature to date. It is assumed that it is possible to measure both species separately, but only one control input is available, namely the dilution rate. A system analysis allows for the determination of infinitely many, but uniquely determined steady-states that are used to derive nonlinear input-output dynamics via the relation of hyperbolic partial differential equations to integral delay equations. A model inversion yields a feedforward control to control the two different outputs, which are a weighted integral over the population density. This results in a restriction in the choice of reference trajectories due to the undercount of inputs. Simulations show the great potential that can be achieved with model-based feedforward control in the context of population systems.

Stability and convergence of Galerkin schemes for parabolic equations with application to Kolmogorov pricing equations in time-inhomogeneous Lévy models

Two essential quantities for the analysis of approximation schemes of evolution equations are stability and convergence. We derive stability and convergence of fully discrete approximation schemes of solutions to linear parabolic evolution equations governed by time dependent coercive operators. We consider abstract Galerkin approximations in space combined with theta-schemes in time. The level of generality of our analysis comprises both a large class of time-dependent operators and a large choice of approximating Galerkin spaces. In particular the results apply to partial integro differential equations for option pricing in time-inhomogeneous Levy models and allows for a large variety of option types and models. The derivation builds on the strong foundation laid out by von Petersdorff and Schwab (2003) who provide the respective results for the time-homogeneous case. We discuss the assumptions in the context of option pricing.

Distributed control of linear partial integro-differential equations based on the input-output linearization approach

Distributed control of linear partial integro-differential equations based on the input-output linearization approach

Numerical Solution of Fractional Partial Integro-Differential Equations Via Two-Dimensional Fractional Muntz-Legendre Polynomials

Numerical Solution of Fractional Partial Integro-Differential Equations Via Two-Dimensional Fractional Muntz-Legendre Polynomials

Feedback control of stochastic gene switches using PIDE models

Achieving control of gene regulatory circuits is one of the goals of synthetic biology, as a way to regulate cellular functions for useful purposes (in biomedical, environmental or industrial applications). The inherent stochastic nature of gene expression makes it challenging to control the behavior of gene regulatory networks, and increasing efforts are being devoted in the field to address different control problems.In this work, we combine the efficient modeling of stochastic gene regulatory networks by means of Partial Integro-Differential Equations with feedback control, in order to keep protein levels at the target (pre-defined) stationary probability distribution. In particular, we achieve the closedloop stabilization of bi-modal toggle-switches in the stochastic regime within the region of low probability (around the minimum located between the two modes of the uncontrolled system).

A comparative study on some semi-analytical methods for the solutions of fractional partial integro-differential equations

A comparative study on some semi-analytical methods for the solutions of fractional partial integro-differential equations

First passage times as a measure of hysteresis in stochastic gene regulatory circuits

In the context of phenotype switching and cell fate determination, numerousexperimental studies report hysteresis, despite the fact that the (forward) Chemical Master Equation governing the inherently stochastic underlying gene regulatory networks has a unique steady state (precluding memory effects and hysteresis). In previous works, we demonstrate thathysteresis is a transient phenomenon in systems far from the thermodynamic limit, using the convergence rates of the partial integro-differential equation associated to the forward master equation governing the stochastic process. Here, we make use of the backward master equationto quantify hysteresis and irreversibility based on First Passage Times..First, we derive the backward master equation for a gene regulatory network with protein production in bursts. Solving this equation, we obtain the probability distributions of the first times to reach some fixed final state from one starting state. The mean first passage time provides a measure to quantify how hysteresis and irreversibility in gene regulation at the singlecell level are transient effects that vanish at steady state. In addition, we provide a theoretical basis that reconciles phenotype coexistence and prevalence far from the thermodynamic limit. In fact, we substitute the notion of pseudo-potential (the so-called Waddington landscape) by a time evolving landscape built upon the Chemical Master Equation (CME) in which phenotypes,rather than prevail, persist with different intensities.

Leaderless Consensus Control of Nonlinear PIDE-Type Multi-Agent Systems With Time Delays

This paper studies leaderless consensus of semi-linear parabolic partial integro-differential equations based multi-agent systems (PIDEMASs) with time delays. Making use of the information interaction and coordination among the neighboring agents, consensus control of the leaderless PIDEMAS is constructed. Consensus of the leaderless PIDEMAS is analyzed by using a Lyapunov approach. Dealing with time-invariant delays and time-varying delays, two sufficient conditions for consensus of the leaderless PIDEMAS are respectively obtained in terms of LMIs. Two examples illustrate the effectiveness of developed theoretical results.

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

<p style='text-indent:20px;'>The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the geometric step option pricing literature. First, we provide symmetry and duality relations and derive various characterizations for both European-type and American-type geometric double barrier step options. In particular, we are able to obtain a jump-diffusion disentanglement for the early exercise premium of American-type geometric double barrier step contracts and its maturity-randomized equivalent as well as to characterize the diffusion and jump contributions to these early exercise premiums separately by means of partial integro-differential equations and ordinary integro-differential equations. As an application of our characterizations, we derive semi-analytical pricing results for (regular) European-type and American-type geometric down-and-out step call options under hyper-exponential jump-diffusion models. Lastly, we use the latter results to discuss the early exercise structure of geometric step options once jumps are added and to subsequently provide an analysis of the impact of jumps on the price and hedging parameters of (European-type and American-type) geometric step contracts.</p>

Experimental and theoretical investigation of CVT rubber belt vibrations

Abstract This article contains the results of experimental tests of vibrations of a continuously variable transmission (CVT) belt transmission driven by a two-stroke internal combustion engine. The measurements were made with the use of a high-speed camera, which allowed to obtain results impossible to obtain with other measurement methods. The nonlinear integro-partial differential equation of vibrations of the moving belt is given. Based on a simplified linear equation, the observed effects on the amplitude–frequency characteristics obtained from the measurements are explained. An approximate formula is given that allows for determining the transmission belt velocities for which resonances occur.

Application of Triple Shehu Transformation for solving Volterra Integro-Partial Differential Equation

This paper depends on some concepts related to the triple Shehu transform; and how its various properties can be used to solve three-dimensional linear Volterra Integro-Partial Differential Equations; in terms of obtaining the exact solutions. Some examples will be studied to support the effectiveness of this transform.

Numerical valuation of European and American options under Merton's model

In this paper, a new combination of time and spatial discretization is proposed for a partial integro-differential equation (PIDE) arising in the valuation of European options under Merton's model. We first present a high-order compact (HOC) difference scheme in space based on a uniform mesh to obtain a highly accurate result, and the discontinuous Galerkin (DG) finite element method in time is introduced that can deal with the loss of the time analyticity. A penalty method is proposed for a partial integro-differential complementarity problem arising in the valuation of the American put option. Numerical experiments are performed to verify the accuracy and efficiency of the proposed method.

A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations

A novel approach for solving multi-term time fractional Volterra–Fredholm partial integro-differential equations

Uniqueness of the partial integro-differential equations

We study the uniqueness of solutions for certain partial integro-differential equations with the initial conditions in a Banach space. The results derived are new and based on Babenko’s approach, convolution and Banach’s contraction principle. We also include several examples for the illustration of main theorems.

Evolutionary dynamics in an SI epidemic model with phenotype-structured susceptible compartment.

We present an SI epidemic model whereby a continuous structuring variable captures variability in proliferative potential and resistance to infection among susceptible individuals. The occurrence of heritable, spontaneous changes in these phenotypic characteristics and the presence of a fitness trade-off between resistance to infection and proliferative potential are explicitly incorporated into the model. The model comprises an ordinary differential equation for the number of infected individuals that is coupled with a partial integrodifferential equation for the population density function of susceptible individuals through an integral term. The expression for the basic reproduction number [Formula: see text] is derived, the disease-free equilibrium and endemic equilibrium of the model are characterised and a threshold theorem involving [Formula: see text] is proved. Analytical results are integrated with the results of numerical simulations of a calibrated version of the model based on the results of artificial selection experiments in a host-parasite system. The results of our mathematical study disentangle the impact of different evolutionary parameters on the spread of infectious diseases and the consequent phenotypic adaption of susceptible individuals. In particular, these results provide a theoretical basis for the observation that infectious diseases exerting stronger selective pressures on susceptible individuals and being characterised by higher infection rates are more likely to spread. Moreover, our results indicate that heritable, spontaneous phenotypic changes in proliferative potential and resistance to infection can either promote or prevent the spread of infectious diseases depending on the strength of selection acting on susceptible individuals prior to infection. Finally, we demonstrate that, when an endemic equilibrium is established, higher levels of resistance to infection and lower degrees of phenotypic heterogeneity among susceptible individuals are to be expected in the presence of infections which are characterised by lower rates of death and exert stronger selective pressures.

Application of finite element method to General Dynamic Equation of Aerosols – Comparison with classical numerical approximations

This paper focuses on the numerical approximation of the general dynamic equation of aerosols (GDE), an integro-partial differential equation, which models the temporal evolution of aerosol number distributions. More specifically, we study the applicability of the finite element method (FEM) to numerically approximate the GDE at different conditions. In addition to applying the conventional Galerkin FEM to the GDE, we also introduce its extension, the Petrov–Galerkin FEM to improve the stability of the approximation. The FEM approximations are compared against the so-called sectional method, based on the finite difference approximation, which is commonly used for the numerical approximation of the GDE. The FEM schemes and the sectional method are validated with series of numerical simulations, where an accurate solution of GDE is available. In these simulations, we consider special cases of pure condensation and coagulation, as well as cases including both of these processes. The accurate solutions – to which numerical approximations are compared – consist of either the analytical solutions or an accurate solution to the discrete GDE, where the particle size range is discretized into intervals of the size of a monomer, and the condensation process is modeled as an interaction between aerosol particles and monomers. The results of this paper demonstrate that while in the solution of the coagulation problem, the conventional sectional method is more efficient than the FEM, in cases of the condensation equation and the condensation-dominated GDE, the FE approximations outperform the sectional method – yielding a desired level of accuracy of the solution using less discretization points and in computation time which can be even orders of magnitude lower than with the sectional method.

Compensation of spatially-varying state delay for a first-order hyperbolic PIDE using boundary control

This paper develops a spatially-varying delay-compensated control for a first-order hyperbolic partial integro differential equation (PIDE). By introducing a 2-dimensional (2-D) transport partial differential equation (PDE) with a spatially-dependent propagation speed, we transform the delay from the time domain to space domain and which results an equivalent cascade system composed of a PIDE and a transport PDE. The control design is based on the backstepping method with a novel transformation which contains the weighted integration of the shifted state of the transport PDE (actuator state) besides the weighted integration of the PIDE state and the weighted double integration of the actuator state. The kernel equations and the inverse kernel equations are proved to be well-posedness by the successive approximation method, respectively. The finite-time stability of the closed-loop system is established and finally, a simulation example is provided to illustrate the effectiveness of the proposed delay-compensated controller.

Risk-sensitive zero-sum stochastic differential game for jump–diffusions

This article is concerned with the infinite horizon risk-sensitive zero-sum stochastic differential game problem for a class of jump–diffusions controlled through the drift, and driven by a compensated Poisson process and a Wiener process. Under certain geometric stability assumption on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov strategies. We obtain our result by exploiting the stochastic representation of the principal eigenfunction of the associated Hamilton–Jacobi–Isaac (HJI) equation, which is a semilinear integro-partial differential equation.

An Efficient Numerical Scheme for the Solution of a Stochastic Volatility Model Including Contemporaneous Jumps in Finance

The model of stochastic volatility with contemporaneous jumps is written for pricing under a partial integro-differential equation (PIDE) having a double integral and a nonsmooth initial value. To tackle this problem, first, a new radial basis function (RBF) as a convex combination of two known RBFs is given. Second, the weighting coefficients of the RBF generated finite difference (FD) method are contributed and the associated error equations are derived. To deal with the integral part, the new idea is to apply an estimate for the unknown function for every cell and do an integration of the density function. The contributed approach is competitive and reduces both the calculational efforts and elapsed time.