Nonlinear Integro-differential Equations Research Articles

Sampling constrained stochastic trajectories using Brownian bridges.

We present a new method to sample conditioned trajectories of a system evolving under Langevin dynamics based on Brownian bridges. The trajectories are conditioned to end at a certain point (or in a certain region) in space. The bridge equations can be recast exactly in the form of a non-linear stochastic integro-differential equation. This equation can be very well approximated when the trajectories are closely bundled together in space, i.e., at low temperature, or for transition paths. The approximate equation can be solved iteratively using a fixed point method. We discuss how to choose the initial trajectories and show some examples of the performance of this method on some simple problems. This method allows us to generate conditioned trajectories with a high accuracy.

Solving the Boundary Problem of a Non-Stationary Equation Transfer of Radiation and Particles for a Semi-Infinite Medium

The boundary problem of non-stationary radiative and particle transfer equation for a semi-infinite medium with an arbitrary single scattering law is considered. This problem comes to finding of a path length distribution function for particles in a turbid medium. A non-linear integro-differential equation for path length distribution function in the case of non-stationary multiple scattering in a semi-infinite medium with an anisotropic scattering law is found by means of invariant embedding. With the help of the discrete ordinates method, matrix non-linear differential equations are deduced that are solved by formulae of backward differentiation and matrix methods for solution of the Lyapunov equation. The computing results are verified by the Monte Carlo method for the path length distribution function for photons backscattered from a drop WC1 cloud and elastically scattered electrons backscattered from a solid semi-infinite target.

Quasinormal Forms for Chains of Coupled Logistic Equations with Delay

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

Waves on the surface of a thin liquid film entrained by a turbulent gas flow in a narrow channel

The paper considers the problem of a joint flow in a narrow vertical channel of a turbulent gas flow and a wave film of liquid flowing down its walls. The calculation of tangential stresses on the interface is performed. The components of the Reynolds stress tensor were determined within the framework of the Boussinesq hypothesis. The turbulent viscosity coefficient was taken into account both when calculating the average velocity profile and when considering disturbances caused by wavy interface. The main difference this case and the semi-infinite channel case is observed in the longwave region. Nonlinear waves on a liquid film flowing down under the action of gravity are studied in a known stress field at the interface. For the case of low Reynolds numbers of the liquid (Re ∼1), the problem is reduced to a nonlinear integro-differential equation for the deviation of the layer thickness from the unperturbed level. For this equation, the character of branching from the unperturbed solution of weakly nonlinear steady-state traveling solutions, whose wave numbers lie in the neighborhood of neutral points, is studied. A numerical study of the evolution of periodic perturbations, whose wave numbers noticeably differ from neutral wave numbers, has been carried out. Several characteristic scenarios for the development of such disturbances are presented.

On Nonlinear Multi-Term Fractional Integro-Differential Equations with Anti-Periodic Boundary Conditions

On Nonlinear Multi-Term Fractional Integro-Differential Equations with Anti-Periodic Boundary Conditions

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

A Compact Difference Scheme on Graded Meshes for the Nonlinear Fractional Integro-differential Equation with Non-smooth Solutions

Optimal error estimates of Galerkin method for a nonlinear parabolic integro-differential equation

In this paper, the error analysis of Galerkin finite element method (FEM) is investigated for a nonlinear parabolic integro-differential equation in two dimensions. By skillfully and rigorously manipulating the nonlinear term, optimal error estimates in L∞(L2(Ω)) and L∞(H1(Ω)) are obtained for a linearized backward Euler fully-discrete scheme, which improves the suboptimal approximation in L∞(L2(Ω)) in the previous literature. Finally, some numerical results are provided to verify the theoretical findings.

Stochastic response determination of U-OWC energy harvesters: a statistical linearization solution treatment accounting for intermittent wave excitation

A novel statistical linearization technique is developed for determining approximately the response statistics and the power output of U-Oscillating Water Column (U-OWC) energy harvesting systems. In this regard, first, the governing equations are derived by employing the unsteady Bernoulli equation. Note that the intermittent, i.e., non-stationary, nature of the wave excitation, occurring in severe sea states due to uncovering of the U-OWC inlet, is explicitly accounted for in the herein proposed model. This is done by multiplying the excitation process with a Heaviside function dependent on the instantaneous free surface displacement. Next, the resulting coupled system of nonlinear integro-differential stochastic equations is solved approximately by relying on a statistical linearization technique. Specifically, the original system of nonlinear equations is replaced by an equivalent linear one, whose parameters and response first- and second-order statistics are obtained by minimizing the mean square error between the two systems. A significant novel aspect of the technique relates to the fact that the Heaviside function is replaced in the equivalent linear system by an “equivalent excitation” coefficient to be determined as part of the statistical linearization solution scheme. Further, compared with other relevant solution schemes adopted in earlier research efforts in the literature, it is shown that the developed technique can be construed as a direct generalization that exhibits an enhanced accuracy degree. The U-OWC installed in the Civitavecchia harbor (Rome, Italy) is considered as an illustrative numerical example, where the reliability of the approximate technique is demonstrated by comparisons with pertinent Monte Carlo simulation data.

Crank–Nicolson extrapolation and finite element method for the Oldroyd fluid with the midpoint rule

An efficient method is examined for the Oldroyd fluid which belongs to nonlinear integro-differential equations. This approach is approximately discretized by the FEM (i.e. finite element method) in space level, and a second-order Crank–Nicolson extrapolation in time. It reduces nonlinear equations to linear equations. Thus can greatly increase the computational efficiency. In order to approximate the integral term, we apply the “midpoint rule” approach. The error of the integral term to the midpoint rule is determined. We establish that the method is unconditionally stable and convergent. The L2-optimal error estimate with second order accuracy in time and space level is obtained without imposing any restriction on time-step size, while all previous works of higher-order scheme require certain time-step conditions. Moreover, corresponding numerical experiments are shown to demonstrate the accuracy, the unconditional stability and convergence of this method for the Oldroyd fluid. The structure is also suitable for the Navier–Stokes equations (i.e. ρ=0 in the Oldroyd fluid).

The two-phase Stefan problem with anomalous diffusion

The non-local in space two-phase Stefan problem (a prototype in phase change problems) can be formulated via a singular nonlinear parabolic integro-differential equation which admits a unique weak solution. This formulation makes Stefan problem to be part of the General Filtration Problems; a class which includes the Porous Medium Equation. In this work, we prove that the weak solutions to both Stefan and Porous Media problems are continuous.

Existence Results for a Multipoint Fractional Boundary Value Problem in the Fractional Derivative Banach Space

We study a class of nonlinear implicit fractional differential equations subject to nonlocal boundary conditions expressed in terms of nonlinear integro-differential equations. Using the Krasnosel’skii fixed-point theorem we prove, via the Kolmogorov–Riesz criteria, the existence of solutions. The existence results are established in a specific fractional derivative Banach space and they are illustrated by two numerical examples.

The existence and compactness of the set of solutions for a 2-order nonlinear integrodifferential equation in N variables in a Banach space

The existence and compactness of the set of solutions for a 2-order nonlinear integrodifferential equation in N variables in a Banach space

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels

Existence results for a nonlinear coupled radiative–conductive heat transfer system

In this paper, we present existence results for a one-dimensional coupled system of nonlinear integro-differential equations on Lp-spaces with 1≤p<∞. This system describes the steady-state combined radiative–conductive heat transfer. We reformulate the full coupled system as a fixed point problem. This formulation is applied to prove existence results for the radiation intensity and the temperature fields. We treat separately the cases p=1 and p>1. The analysis for p>1 uses the Krasnosel’skii fixed point theorem, while the existence results in the case p=1 are obtained via a new variant of that theorem for the weak topology established in Latrach et al. (2006).

$$W^{\sigma ,p}$$ A priori estimates for fully nonlinear integro-differential equations

$W^{\sigma,p}$ estimates are studied for a class of fully nonlinear integro-differential equations of order $\sigma$, which are analogues of $W^{2,p}$ estimates by Caffarelli. We also present Aleksandrov-Bakelman-Pucci maximum principles, which are improvements of estimates proved by Guillen-Schwab, depending only on $L^p$ norms of inhomogeneous terms.

On the Boundary Value Problem of Nonlinear Fractional Integro-Differential Equations

Using Banach’s contractive principle and the Laray–Schauder fixed point theorem, we study the uniqueness and existence of solutions to a nonlinear two-term fractional integro-differential equation with the boundary condition based on Babenko’s approach and the Mittag–Leffler function. The current work also corrects major errors in the published paper dealing with a one-term differential equation. Furthermore, we provide examples to illustrate the application of our main theorems.

Study on existence and data dependence results for fractional order differential equations

In this paper, we examine the multi-derivative nonlinear fractional integro-differential equations involving Atangana-Baleanu fractional derivative of Riemann-Liouville sense. We study the elementary results about the existence and difference solution on different data, based on T. R. Prabhakar fractional integral operator εσ,η,ϑ;c+γ involving generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and Gronwall-Bellman inequality.2010 Subject Classification: 34A60,33E12,34G20.

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.

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

An approach based on fractional-order Lagrange polynomials for the numerical approximation of fractional order non-linear Volterra-Fredholm integro-differential equations

Existence and Uniqueness of Solutions for Nonlinear Fractional Integro-Differential Equations with Nonlocal Boundary Conditions

In this paper, the existence and uniqueness of solutions of fractional integro-differential equations with nonlocal boundary conditions is investigated. We establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle.