Integro-differential Equations Research Articles

Mixed virtual element method for integro-differential equations of parabolic type

Mixed virtual element method for integro-differential equations of parabolic type

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Solving higher-order nonlinear Volterra integro-differential equations using two discretization methods

Trigonometric weighted generalized convolution operator associated with Fourier cosine–sine and Kontorovich–Lebedev transformations

ABSTRACT The main objective of this work is to introduce the generalized convolution with trigonometric weighted γ = sin ⁡ y involving the Fourier cosine–sine and Kontorovich–Lebedev transforms, and to study its fundamental results. We establish the boundedness properties in a two-parametric family of Lebesgue spaces for this convolution operator. Norm estimation in the weighted L p space is obtained and applications of the corresponding class of convolution integro-differential equations are discussed. The conditions for the solvability of these equations in L 1 space are also founded.

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

Existence of positive solutions for a p-Schrödinger–Kirchhoff integro-differential equation with critical growth

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

On Some Impulsive Fractional Integro-Differential Equation with Anti-Periodic Conditions

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results.

Forest fire spreading: A nonlinear stochastic model continuous in space and time

AbstractForest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed‐point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

New investigation on controllability of sobolev-type Volterra-Fredholm functional integro-differential equation with non-local condition

In this article, we analyse the controllability criteria on sobolev-type Volterra-Fredholm functional Integro-differential Equations (SVFIDE) via Atangana–Baleanu Caputo (ABC) derivative. The primary outcomes was established by utlising the concepts on the measure of noncompactness, semigroup theory, the contraction principle and Darbo fixed point technique with nonlocal condition. An illustrative example to demonstrate the analytical result is provided at the end.

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

Semilinear multi-term fractional in time diffusion with memory

In this study, the initial-boundary value problems to semilinear integro-differential equations with multi-term fractional Caputo derivatives are analyzed. A particular case of these equations models oxygen diffusion through capillaries. Under proper requirements on the given data in the model, the classical and strong solvability of these problems for any finite time interval [0, T] are proved via so-called continuation method. The key point in this approach is finding suitable a priori estimates of a solution in the fractional Hölder and Sobolev spaces.

Real option pricing under the regime-switching model with jumps on a finite time horizon

We consider an irreversible investment decision problem on a finite time horizon where an instantaneous cash flow process of a firm follows a regime-switching jump-diffusion (RSJD) model. The value of a project can be derived from a partial integro-differential equation (PIDE) and then we obtain a closed-form solution of the PIDE. It is proved that the value of the project converges to the solution on an infinite time horizon problem as the lifetime of the project tends to infinity. Moreover, the value function of a real option with a finite expiration date can be evaluated by solving a linear complementarity problem (LCP) under the RSJD model. We apply a 2-step backward differentiation formula (BDF2 method) combined with an operator splitting method to solve the LCP and then it has the second-order convergence rate in the discrete ℓ2-norm with respect to the time and spatial variables. The stability of the proposed method to solve the LCP is also proved in the discrete ℓ2-norm. Finally, a variety of numerical experiments are carried out to determine the optimal investment time and the numerical results on the finite time horizon are analyzed.

Numerical Solution on Neutral Delay Volterra Integro-Differential Equation

Numerical Solution on Neutral Delay Volterra Integro-Differential Equation

Finite Difference Method for Solving Parabolic-Type Integro-Differential Equations in Multidimensional Domain with Nonhomogeneous First-Order Boundary Conditions

Introduction. We investigate a multidimensional (in terms of spatial variables) parabolic-type integro-differential equation with nonhomogeneous first-order boundary conditions. The locally one-dimensional finite difference scheme developed herein can be applied to solve various applied problems leading to multidimensional parabolic-type integro-differential equations. Examples include mathematical modelling of cloud processes, addressing the issue of active intervention in convective clouds to prevent hail and artificially enhance precipitation, as well as describing the droplet mass distribution function due to microphysical processes such as condensation, coagulation, fragmentation, and freezing of droplets in convective clouds.Materials and Methods. In this study, an approximate solution to the initial-boundary value problem is constructed using the locally one-dimensional scheme of A.A. Samarsky with a specified order of approximation О(h2 +τ). The primary research method employed is the method of energy inequalities.Results. A priori estimates have been obtained in the discrete interpretation, from which uniqueness, stability, and convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem follow, with a convergence rate equal to the order of approximation of the difference scheme.Discussion and Conclusions. The research findings can be utilized for further development of boundary value problem theory for parabolic equations with variable coefficients. Additionally, they may find applications in the fields of difference scheme theory, computational mathematics, and numerical modelling.

Numerical solutions for nonlinear mixed integro-differential equation and the neutron transport equation

In this work, I will discuss the solution of the new in a nonlinear integro-differential equation (NI-DE) form. The NI-DE is established from the nuclear transport equation, which is a linearized derivative of the equation developed by Boltzmann for the kinetic theory of gases. In this study, I will introduce the basic equations and boundary conditions for the formulation of the problem. Then, the NI-DE is established from the planar geometry problem of the neutron transport equation. In addition, I will prove the existence of a unique solution to the problem. In addition, I will use the numerical method to obtain a system of NI-DEs. Therefore, I will prove, under certain conditions, the existence of a unique solution to this system. Finally, I will present two different effective methods to solve the problem numerically, and I will discuss the results.

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Error Analysis of Fractional Collocation Methods for Volterra Integro-Differential Equations with Noncompact Operators

Effects of buoyancy on the dispersion of drugs released intrathecally in the spinal canal.

This paper investigates the transport of drugs delivered by direct injection into the cerebrospinal fluid (CSF) that fills the intrathecal space surrounding the spinal cord. Because of the small drug diffusivity, the dispersion of neutrally buoyant drugs has been shown in previous work to rely mainly on the mean Lagrangian flow associated with the CSF oscillatory motion. Attention is given here to effects of buoyancy, arising when the drug density differs from the CSF density. For the typical density differences found in applications, the associated Richardson number is shown to be of order unity, so that the Lagrangian drift includes a buoyancy-induced component that depends on the spatial distribution of the drug, resulting in a slowly evolving cycle-averaged flow problem that can be analysed with two-time scale methods. The asymptotic analysis leads to a nonlinear integro-differential equation for the spatiotemporal solute evolution that describes accurately drug dispersion at a fraction of the cost involved in direct numerical simulations of the oscillatory flow. The model equation is used to predict drug dispersion of positively and negatively buoyant drugs in an anatomically correct spinal canal, with separate attention given to drug delivery via bolus injection and constant infusion.