Partial Integro-differential Equations Research Articles

Qualitative Properties of Space-Dependent SIR Models with Constant Delay and Their Numerical Solutions

Abstract In this article, a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that, by choosing the time step to be sufficiently small, the schemes preserve the qualitative properties of the original continuous model. Finally, some numerical experiments are presented that confirm the aforementioned theoretical results.

Pell Collocation Method for Solving the Nonlinear Time–Fractional Partial Integro–Differential Equation with a Weakly Singular Kernel

This article focuses on finding the numerical solution of the nonlinear time–fractional partial integro–differential equation. For this purpose, we use the operational matrices based on Pell polynomials to approximate fractional Caputo derivative, nonlinear, and integro–differential terms; and by collocation points, we transform the problem to a system of nonlinear equations. This nonlinear system can be solved by the fsolve command in Matlab. The method’s stability and convergence have been studied. Also included are five numerical examples to demonstrate the veracity of the suggested strategy.

Simulation of the crystallization processes by population balance model using a linear separation method

Abstract Crystallization problem is one of the popular problems in wide area of science. The first principles are not used to design a crystallizer in which complicated processes include nucleation, crystal growth, attrition and agglomeration of crystals. It is modeled by the population balance model, which is one of the important models of mathematical biology and engineering, is a nonlinear partial integro-differential equation and examines the exchange of particles and the production of new particles in a system of particles. For the crystallization problem, one-dimensional and multi-dimensional models are considered and semi-analytical solutions are obtained via the linear separation method.

A Deep Neural Network Approach to Solving for Seal’s Type Partial Integro-Differential Equation

In this paper, we study the problem of solving Seal’s type partial integro-differential equations (PIDEs) for the classical compound Poisson risk model. A data-driven deep neural network (DNN) method is proposed to calculate finite-time survival probability, and an alternative scheme is also investigated when claim payments are exponentially distributed. The DNN method is then extended to the numerical solution of generalized PIDEs. Numerical approximation results under different claim distributions are given, which show that the proposed scheme can obtain accurate results under different claim distributions.

Collocation Approach Based on an Extended Cubic B -Spline for a Second-Order Volterra Partial Integrodifferential Equation

This paper focuses on an efficient spline-based numerical technique for numerically addressing a second-order Volterra partial integrodifferential equation. The time derivative is discretized using a finite difference scheme, while the space derivative is approximated using the extended cubic B -spline basis. The scheme is also tested for stability study to ensure that the errors do not accumulate. The convergence of the proposed scheme is also investigated. The scheme’s key benefit is that the approximate solution is produced as a smooth piecewise continuous function allowing us to approximate the solution at any location in the domain. Numerical study is performed, and the comparison of results is made to previously reported results in the literature to show the efficiency of the suggested scheme.

On the Existence of Self-Similar Solutions in the Thermostatted Kinetic Theory with Unbounded Activity Domain

This paper is devoted to the mathematical analysis of a spatially homogeneous thermostatted kinetic theory framework with an unbounded activity domain. The framework consists of a partial integro-differential equation with quadratic nonlinearity where the domain of the activity variable is the whole real line. Specifically the mathematical analysis refers firstly to the existence and uniqueness of the solution for the related initial boundary value problem; Secondly the investigations are addressed to the existence of a class of self-similar solutions by employing the Fourier transform method. In particular the main result is obtained for a nonconstant interaction rate and a nonconstant force field. Conclusions and perspectives are discussed in the last section of the paper.

Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation

In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.

Existence results for some partial stochastic integrodifferential equations with nonlocal conditions in Hilbert spaces

Existence results for some partial stochastic integrodifferential equations with nonlocal conditions in Hilbert spaces

Output regulation for a first-order hyperbolic PIDE with state and sensor delays

Considering disturbances within domain and at the boundary, a backstepping-based output boundary regulator design is developed for a class of first-order linear hyperbolic partial integro-differential equation (PIDE) in the presence of state and sensor delays. The delays are represented by two transport PDEs, which results in an extended spatial domain where the hyperbolic PIDE, transport PDEs and ordinary differential equation (ODE) are in cascade. The ODE is a finite-dimensional signal model describing exogenous signals. First, a state feedback regulator is realized to achieve a finite time stability by applying an affine Volterra integral transformation. Then, an output regulator is developed on the basis of the nominal plant transfer behavior, which results in an exponential stability. Numerical examples illustrate the performance of the proposed regulators.

On the behaviour solutions of fractional and partial integro differential heat equations and its numerical solutions

Abstract In this paper, the semi-group method is used to discuss the existence and uniqueness of solutions for fractional and partial integro differential equations (F-PIDEs) of heat type in Banach space E. In addition, the stability of the solutions for F-PIDEs are discussed. Moreover, the Adomian decomposition method (ADM) is used to obtain the solutions numerically. Finally, numerical results of each case are obtained, and the difference of results between the fractional partial integro differential equation and partial integro differential equation are explained. Furthermore, the error is computed in each case.

A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel

The main aim of this paper is to solve the two-dimensional nonlinear fractional partial integro-differential equation (PIDE) with a weakly singular kernel by using the time two-grid finite difference (FD) algorithm. The second-order backward difference formula (BDF) and L1 scheme are used in time. The time two-grid algorithm is constructed to improve the solving efficiency of nonlinear systems. The Newton iteration is used to solve nonlinear discrete system on the coarse grid, and then we apply Lagrangian linear interpolation to attain the function value used in constructing the difference scheme on the fine grid. The second-order finite difference method (FDM) is used in space. The unconditional stability and convergence are attained for the two-grid fully discrete system. Numerical experiments show that the used CPU time for the presented two-grid numerical algorithm is lower than the general finite difference method for solving the nonlinear system.

Nonlinear electro-elastic dynamics of a hub–cantilever bimorph rotor structure

Dynamics of a rotating mini hub–bimorph structure excited by periodic torque supplied to the hub is studied in this paper. The mathematical model of the system is formulated by means of the extended Hamilton’s principle. In the analysis the nonlinear constitutive behaviour of the piezoceramic material is adopted based on experimental results published in literature. Moreover, nonlinear inertia effects related to the system rotation are taken into account. The derived system of three integro-partial differential equations represents the electro-mechanical behaviour of the beam (transverse displacement and transducer output voltage) and the hub rotation coordinate. The state equations are discretized and solved numerically around the first resonance zone for the system excited by a periodic torque supplied to the hub. The obtained response curves reveal strong softening behaviour with additional unstable solutions identified out of the resonance zones. Within the framework of the performed numerical simulations the impact of torque excitation amplitude, torque mean value, hub inertia and electrical load resistance on system behaviour is discussed. Finally, the responses of the system with parallel vs series connected transducers are compared.

Comparison of Two Hybrid Functions for Numerical Solution of Nonlinear Mixed Partial Integro-Differential Equations

Comparison of Two Hybrid Functions for Numerical Solution of Nonlinear Mixed Partial Integro-Differential Equations

Nutritionally rich wildflower patches adjacent to nutritionally deficient crops significantly increase pollination services

The balance between nutrition and energy needs has an important impact on the spatial distribution of foraging animals. In the present paper, we focus on the case of bumble bees moving around a landscape in search of pollen (to meet nutritional needs) and nectar (to meet energy needs). Depending on the colony demands, bumble bees can concentrate their foraging effort towards either pollen rich flower species or nectar rich flower species. This behaviour allows us to establish a strategy – a spatial landscape design – which can maximize the pollination services of crops that are nutritionally deficient for pollinators by adding nutritionally rich wildflower patches. To do this, we formulate a mathematical partial integro-differential equation model to predict the spatial distribution of foraging bumble bees. We use our model to predict the location, composition and quantity of the wildflower patches adjacent to crop fields that will be most beneficial for crop pollination services. Our results show that relatively small quantities of wildflowers in specific locations with respect to the nest sites and the crop can have a positive impact on pollination services when the composition (i.e., pollen to nectar ratio) of the added wildflowers is significantly different from the composition of the existing crop flowers.

The Solution of the Nonlinear Mixed Partial Integro-differential Equation via Two-Dimensional Hybrid Functions

The Solution of the Nonlinear Mixed Partial Integro-differential Equation via Two-Dimensional Hybrid Functions

Approximate solutions of aggregation and breakage population balance equations

The work aims to present a semi-analytical approach based on the homotopy analysis method for finding closed-form solutions and approximate solutions to aggregation and breakage population balance equations. The population balance equations are specific integro-partial differential equations. In this work, we first transform both the aggregation and breakage models into integral equations. Then the resulting integral equations are solved by the homotopy analysis method (HAM) to get the series solution, which in particular cases eventually converges to the exact solution. The semi-analytical solutions for various benchmark aggregation kernels like the Ruckenstein/Pulvermacher kernel and breakage kernels such as the Austin kernel are provided using HAM.

Local-density dependent Markov processes on graphons with epidemiological applications

We investigate local-density dependent Markov processes on a class of large graphs sampled from a graphon, where the transition rates of the vertices are influenced by the states of their neighbors. We show that as the average degree converges to infinity (faster than a given threshold), the evolution of the process in the transient regime converges to the solution of a set of non-local integro-partial differential equations depending on the limit graphon. We also provide rigorous derivation for the epidemic threshold in the case of the Susceptible–Infected–Susceptible (SIS) process on such graphons.

Representing and Analyzing the Dynamics of an Agent-Based Adaptive Social Network Model with Partial Integro-Differential Equations

We formulated and analyzed a set of partial integro-differential equations that capture the dynamics of our adaptive network model of social fragmentation involving behavioral diversity of agents. Previous results showed that, if the agents’ cultural tolerance levels were diversified, the social network could remain connected while maintaining cultural diversity. Here we converted the original agent-based model into a continuous equation-based one so we can gain more theoretical insight into the model dynamics. We restricted the node states to 1-D continuous values and assumed the network size was very large. As a result, we represented the whole system as a set of partial integro-differential equations about two continuous functions: population density and connection density. These functions are defined over both the state and the cultural tolerance of nodes. We conducted numerical integration of the developed equations using a custom-made integrator implemented in Julia. The results obtained were consistent with the simulations of the original agent-based adaptive social network model we previously reported, confirming the robustness of the original finding. Specifically, when the variance of cultural tolerance d is large enough, the population with low d maintains the original clusters of cultures/opinions, while the one with high d tends to come to the center and connect culturally distant groups. Parameter dependence of the model behavior was also revealed through systematic numerical experiments.

Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations

Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations

A numerical scheme for solving a class of time fractional integro-partial differential equations with Caputo–Fabrizio derivative

In this paper, we find the numerical solution of the time-fractional partial integro-differential equation with Caputo–Fabrizio fractional derivative. The problem is discretized by some finite difference schemes in the time direction, and then the Sinc collocation method is applied to the resulting problems in the spatial direction. The convergence and stability of the method are proved and illustrated by three test examples.