Nonlinear Integro-differential Equations Research Articles

Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.

ALGORITHM FOR THE SOLUTION OF NONLINEAR VARIABLE-ORDER PANTOGRAPH FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS USING HAAR METHOD

This research work is related to establish a powerful algorithm for the computation of numerical solution to nonlinear variable order integro-differential equations (VO-IDEs). The adopted procedure is based on the Haar Wavelet Method (HWM) to compute the required numerical solution to the proposed problem. Further, in the considered problem, a proportional-type delay term is involved, which is also known as the pantograph equation. For a physical problem to investigate the computational purposes, we need to first ensure its existence. For this purpose, we utilize classical fixed results given by Banach and Schauder to establish the sufficient conditions for existence of at least one approximate solution to the proposed problem. Two pertinent examples are given, where the error analysis is also recorded.

Snap-through instability of helicoidal composite imperfect beams surrounded by nonlinear elastic foundation

Bernstein polynomials are exploited, for the first time, in investigation of the nonlinear bending and snap-through instability phenomena of geometrically imperfect bioinspired composite beams with helicoidal orientation scheme. The curved bioinspired beam subjected to different types of lateral loading and embedded on three parameters elastic foundations is considered in formulation and analysis. The nonlinear equilibrium equation of the system based on Euler-Bernoulli beam hypothesis, von Kármán nonlinear strain, and initial imperfect curvature is developed. The equilibrium equation portrayed by nonlinear fourth order integro-differential equation is solved by Bernstein polynomials method and closed form solution of the load-deflection relationship is obtained. The accuracy of the present model and solution procedure is examined by comparing our results with previous results. The developed model is exploited to present the influence of lamination scheme, amplitude of initial imperfection and elastic foundation constants on the nonlinear bending and snap-through/reverse snap-through behaviors in detail. The numerical results show that the Bernstein polynomials approach can capture successfully the nonlinear equilibrium path and overcoming limit (turning) points, which usually appear in the nonlinear response of curved structures. The developed model can be used in design and control the snap-through response and open opportunities for this type of structures in many real applications.

Using An accelerated Technique of The Laplace-Adomian Decomposition Method in Solving A class of Non-linear Integro-differential Equations

Using An accelerated Technique of The Laplace-Adomian Decomposition Method in Solving A class of Non-linear Integro-differential Equations

Solutions of Nonlinear Integro-Partial Differential Equations by the Method of G ′ / G , 1 / G

In this article, a special expansion method is implemented in solving nonlinear integro-partial differential equations of 2 + 1 -dimensional using a special expansion method of G ′ / G , 1 / G . We obtained the solutions for 2 + 1 -dimensional nonlinear integro-differential equations in real physical phenomena. The method is applied on 2 + 1 -dimensional space time and solved in three different cases: hyperbolic, trigonometric, and rational functions. The obtained solutions for each result were illustrated by graphical plots using Wolfram Mathematica 9.0 software packages. Furthermore, the obtained results are exactly fit with exact solutions which solves the complicity of finding the solution for nonlinear integro-partial differential equations. Finally, the method is powerful and effective to solve partial differential equations of nonlinear integro form.

General decay stability of backward Euler–Maruyama method for nonlinear stochastic integro-differential equations

The general decay stability of backward Euler–Maruyama(EM) method is discussed for stochastic integro-differential equations(SIDEs) in this paper. Under polynomial growth condition, both continuous and discrete solutions admit high nonlinearity of the underlying equations. Besides, the considered stability namely the general decay stability is a generalized stability including almost sure polynomially stability and almost sure exponentially stability. Making using of the nonnegative semi-martingale convergence theorem, for SIDEs we propose sufficient conditions to obtain the general decay stability of backward EM method.

Analysis of periodic solutions for nonlinear coupled integro-differential systems with variable delays

The objective of this work is the application of Krasnosel'skii's fixed point technique to prove the existence of periodic solutions of a system of coupled nonlinear integro-differential equations with variable delays. An example is given to illustrate this work.

Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity

This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank–Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann–Liouville fractional integral term. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at t=0 so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we employ a standard centered difference formula for the second-order spatial derivative, and the Galerkin method based on piecewise linear test functions is used to approximate the nonlinear convection term. Then we derive the existence and uniqueness of numerical solutions for the proposed implicit difference scheme. Meanwhile, the stability and convergence are proved by means of the energy argument. Furthermore, to demonstrate the effectiveness of the proposed method, we utilize a fixed point iterative algorithm to calculate the discrete scheme. Finally, numerical experiments illustrate the feasibility and efficiency of the proposed scheme, in which numerical results are consistent with our theoretical analysis.

Integrability technique for fluid flow induced deformation of a boundary hair

A solution method for a nonlinear integro-differential equation is proposed and conducted on the problem of a boundary hair exposed to shear flow. While bearing resemblance to the pendulum problem from mechanics, this equation has not been analytically solved until now. We obtain this solution by treating the integral term as a parameter which we later constrain after obtaining the prospective trajectories. Our analytic solution circumvents the difficulties one can have with finite difference along with other numerical approaches and we argue that it could be used as a basis for understanding weakly time-dependent systems.

Solving a Generalized Fractional Nonlinear Integro-Differential Equations via Modified Sumudu Decomposition Transform

The Sumudu decomposition method was used and developed in this paper to find approximate solutions for a general form of fractional integro-differential equation of Volterra and Fredholm types. The Caputo definition was used to deal with fractional derivatives. As the method under consideration depends mainly on writing non-linear terms, which are often found inside the kernel of the integral equation, writing it in the form of Adomian’s polynomials in the well-known way. After applying the Sumudu transformation to both sides of the integral equation, the solution was written in the form of a convergent infinite series whose terms can be alternately calculated. The method was applied to three examples of non-linear integral equations with fractional derivatives. The results that were presented in the form of tables and graphs showed that the method is accurate, effective and highly efficient.

A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels

This paper studies an accurate localized meshless collocation approach for solving two-dimensional nonlinear integro-differential equation (2D-NIDE) with multi-term kernels. The proposed strategy discretizes the unknown solution in two phases. First, the semi-discrete scheme is obtained by using backward Euler finite difference (FD) approach and the first-order convolution quadrature rule for the first order temporal derivative and the Riemann-Liouville (R-L) fractional integral, respectively. Second, the spatial discretization is established by means of the local radial basis function based on partition of unity (LRBF-PU) in the space variable and its partial derivatives. Furthermore, the unconditionally stable result and first-order convergence of the time semi-discrete scheme in L2-norm are proved by the energy method. It is shown that the proposed method is accurate and that the numerical results support the theoretical analysis.

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).