Nonlinear Integro-differential Equations Research Articles

Existence of mild solution to the fractional order impulsive nonlinear control system

In this paper we study the existence of mild solution to the fractional impulsive nonlinear mixedtype integro-differential partial equation with neutral infinite delay and nonlocal conditions in a Banach space. As a prove of this investigation we address some important concepts of work such as fractional calculus, semigroup of operators and Krasnoselskii's fixed point theorem.

NON-AUTONOMOUS FRACTIONAL EVOLUTION EQUATIONS WITH NON-INSTANTANEOUS IMPULSE CONDITIONS OF ORDER (1,2): A CAUCHY PROBLEM

The non-instantaneous condition is utilized in our study through the employment of the Cauchy problem in order to contract a system of nonlinear non-autonomous mixed-type integro-differential (ID) fractional evolution equations in infinite-dimensional Banach spaces. We reveal the existence of new mild solutions in the condition that the nonlinear function modifies approximately suitable, measure of non-compactness (MNC) form and local growth form using evolution classes along with fractional calculus (FC) theory as well as the fixed-point theorem with respect to k-set-contractive operator and MNC standard set. Consequently, as an example, we consider a fractional non-autonomous partial differential equation (PDE) with a homogeneous Dirichlet boundary condition and a non-instantaneous impulse condition. The conclusion of mild solution regarding the uniqueness and existence of a mild solution for a system with a probability density function and evolution classes is drawn with respect to the related domains.

Existence and Controllability of a Class of Non-autonomous Nonlinear Evolution Fractional Integrodifferential Equations with Delay

Existence and Controllability of a Class of Non-autonomous Nonlinear Evolution Fractional Integrodifferential Equations with Delay

On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order

In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new.

Advanced shifted first-kind Chebyshev collocation approach for solving the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel

This research apparatuses an approximate spectral method for the nonlinear time-fractional partial integro-differential equation with a weakly singular kernel (TFPIDE). The main idea of this approach is to set up a new Hilbert space that satisfies the initial and boundary conditions. The new spectral collocation approach is applied to obtain precise numerical approximation using new basis functions based on shifted first-kind Chebyshev polynomials (SCP1K). Furthermore, we support our study by a careful error analysis of the suggested shifted first-kind Chebyshev expansion. The results show that the new approach is very accurate and effective.

VIBRATIONS OF GEOMETRICALLY NONLINEAR VISCOUS-ELASTIC CYLINDRICAL SHELLS INTERACTING WITH AN ELASTIC MEDIUM

This article examines the nonlinear vibrations of a thin-walled structure such as a cylindrical shell interacting with the ground. On the basis of the seismodynamic theory of underground structures, the nature of the movement of the structure is revealed depending on the stiffness coefficient and rheological properties of the soil, as well as on the frequency of external influences. When solving specific problems, seismic waves in the form of a sinusoid are considered. Nonlinear integro-differential equations describing the vibrations of structures laid in the ground are solved approximately in the following sequence:
 - The method of decomposition of displacements in rad by coordinate functions is applied, which are selected depending on the boundary conditions. Using the approximate Bubnov-Galerkin method, the original nonlinear integro-differential equations in partial derivatives of the fourth order are reduced to ordinary equations of the second order;
 - The obtained nonlinear integro-differential equations are solved by the averaging method, as well as numerically.
 - The nature of the change in stress and amplitude of oscillation in time in an elastic and viscoelastic shell at different coefficients of soil stiffness was obtained.

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

On a Class of Nonlinear Integro-Differential Equations

On a Class of Nonlinear Integro-Differential Equations

An Explicit Wavelet Method for Solution of Nonlinear Fractional Wave Equations

An explicit method for solving time fractional wave equations with various nonlinearity is proposed using techniques of Laplace transform and wavelet approximation of functions and their integrals. To construct this method, a generalized Coiflet with N vanishing moments is adopted as the basis function, where N can be any positive even number. As has been shown, convergence order of these approximations can be N. The original fractional wave equation is transformed into a time Volterra-type integro-differential equation associated with a smooth time kernel and spatial derivatives of unknown function by using the technique of Laplace transform. Then, an explicit solution procedure based on the collocation method and the proposed algorithm on integral approximation is established to solve the transformed nonlinear integro-differential equation. Eventually the nonlinear fractional wave equation can be readily and accurately solved. As examples, this method is applied to solve several fractional wave equations with various nonlinearities. Results show that the proposed method can successfully avoid difficulties in the treatment of singularity associated with fractional derivatives. Compared with other existing methods, this method not only has the advantage of high-order accuracy, but it also does not even need to solve the nonlinear spatial system after time discretization to obtain the numerical solution, which significantly reduces the storage and computation cost.

On Some Generalizations of Integral Inequalities in n Independent Variables and Their Applications

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young’s method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities.

Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces

Abstract In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces. We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and the Mönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 2008, 3, 536–546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.

Iterative Method for Solving a Nonlinear Fourth Order Integro-Differential Equation

This study presents the execution of an iterative technique suggested by Temimi and Ansari (TA) method to approximate solutions to a boundary value problem of a 4th-order nonlinear integro-differential equation (4th-ONIDE) of the type Kirchhoff which appears in the study of transverse vibration of hinged shafts. This problem is difficult to solve because there is a non-linear term under the integral sign, however, a number of authors have suggested iterative methods for solving this type of equation. The solution is obtained as a series that merges with the exact solution. Two examples are solved by TA method, the results showed that the proposed technique was effective, accurate, and reliable. Also, for greater reliability, the approximate solutions were compared with the classic Runge-Kutta method (RK4M) where good agreements were observed. For more accuracy the maximum error remainder was found, and the absolute error was computed between the semi-analytical method and the numerical method RK4M. Mathematica® 11 was used as a program for calculations.

A fast numerical solution of the 3D nonlinear tempered fractional integrodifferential equation

AbstractIn this paper, we investigate the numerical solution of the three‐dimensional (3D) nonlinear tempered fractional integrodifferential equation which is subject to the initial and boundary conditions. The backward Euler (BE) method in association with the first‐order convolution quadrature rule is employed to discretize this equation for time, and the Galerkin finite element method is applied for space, which is combined with an alternating direction implicit (ADI) algorithm, in order to reduce the computational cost for solving the three‐dimensional nonlocal problem. Then a fully discrete BE ADI Galerkin finite element scheme can be obtained by linearizing the non‐linear term. Thereafter we prove a positive‐type lemma, from which the stability and convergence of the proposed numerical scheme are derived based on the energy method. Numerical experiments are performed to verify the effectiveness of the proposed approach.

A second-order accurate Crank–Nicolson finite difference method on uniform meshes for nonlinear partial integro-differential equations with weakly singular kernels

A second-order accurate Crank–Nicolson finite difference method on uniform meshes is proposed and analyzed for a class of nonlinear partial integro-differential equations with weakly singular kernels. The first-order time derivative is approximated by using a Crank–Nicolson time-stepping technique, and the singular integral term is treated by a product averaged integration rule, which preserves the positive semi-definite property of the singular integral operator. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative, and a suitable second-order discretization is adopted for the nonlinear convection term. The solvability, stability and convergence of the method are rigorously proved by the discrete energy method, the positive semi-definite property of the associated quadratic form of the method and a perturbation technique. The error estimation shows that the method has the optimal second-order convergence in time and space for non-smooth solutions. Newton’s iterative method and its algorithm implementation are presented to solve the resulting nonlinear system. Numerical results confirm the theoretical convergence result and show the effectiveness of the method.

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

Soliton-like optical pulse in a gain medium with dissipation under conditions of intra-pulse Raman scattering

The propagation of an optical pulse in a nonresonant gain medium with the Raman nonlinearity is studied. A nonlinear integro-differential equation for the envelope of the pulse electric field is derived. The solution of this equation in the form of a superluminal soliton-like pulse is obtained and analyzed.

Buoyancy-modulated Lagrangian drift in wavy-walled vertical channels as a model problem to understand drug dispersion in the spinal canal.

This paper investigates flow and transport in a slender wavy-walled vertical channel subject to a prescribed oscillatory pressure difference between its ends. When the ratio of the stroke length of the pulsatile flow to the channel wavelength is small, the resulting flow velocity is known to include a slow steady-streaming component resulting from the effect of the convective acceleration. Our study considers the additional effect of gravitational forces in configurations with a non-uniform density distribution. Specific attention is given to the slowly evolving buoyancy-modulated flow emerging after the deposition of a finite amount of solute whose density is different from that of the fluid contained in the channel, a relevant problem in connection with drug dispersion in intrathecal drug delivery (ITDD) processes, involving the injection of the drug into the cerebrospinal fluid that fills the spinal canal. It is shown that when the Richardson number is of order unity, the relevant limit in ITDD applications, the resulting buoyancy-induced velocities are comparable to those of steady streaming. As a consequence, the slow time-averaged Lagrangian motion of the fluid, involving the sum of the Stokes drift and the time-averaged Eulerian velocity, is intimately coupled with the transport of the solute, resulting in a slowly evolving problem that can be treated with two-time-scale methods. The asymptotic development leads to a time-averaged, nonlinear integro-differential transport equation that describes the slow dispersion of the solute, thereby circumventing the need to describe the small concentration fluctuations associated with the fast oscillatory motion. The ideas presented here can find application in developing reduced models for future quantitative analyses of drug dispersion in the spinal canal.

Solution estimates and stability tests for nonlinear delay integro-differential equations

In this article, we examine various qualitative features of solutions of a nonlinear delay integro-differential equation. We prove three new theorems which include sufficient conditions on asymptotic stability (AS), integrability, and boundedness of solutions, using a suitable Lyapunov-Krasovskii functional. We present examples that show applications of our results.

"MODEL ACHIEVEMENT FOR IGNITION AND DEVELOPMENT OF STOCHASTIC DISCHARGE CHANNELS IN CONCRETE AND REINFORCED CONCRETE TAKING INTO ACCOUNT THE PROPERTIES OF THE MEDIUM AND THE GEOMETRY OF THE REINFORCING FRAME"

"The article develops a generalized model of electrodischarge action on concrete, which allows consistently describing phases of electro-explosion in condensed media: initiation and development of discharge channels, expansion, generation of shock waves, interaction of waves with the material being processed, deformation and destruction of solid materials. The model is based on a stochastic deterministic approach to the development of instability. processes associated with the distribution of the electric field, when mechanical stresses break. The flow of the process is considered deterministically on the basis of nonlinear integro-differential equations, local processes leading to the growth of the channel and cracks - stochastically. Equations describing the nature of the discharge development, the change of the channel resistance and its expansion are concordantly solved with the transition equations in the scheme of the real pulse generator. Expansion of the channel in the liquid is based on the law of conservation of energy, mass, pulse, equations of wave dynamics and allows to calculate the temporal and amplitude impact of shock waves from the channel on the barriers. The process of electrical breakdown of condensed dielectrics occurs with the development of a stochastically discharge structure, mainly determined by the emergence of highly conductive plasma channels. The development of the channel structure begins in the area of maximum field tension. In this case, the processes of phase transitions of a material directly determine the quantitative and qualitative processes of image of channels. As part of the study of the model, an analysis of the influence of the movement of electro-discharges on the redistribution of the electrostatic field, conditions leading to the direct formation of discharge channels, fracture formation and complete deformation was conducted. Keywords: model, electro-discharge device, stochastic-deterministic approach, electric field, mechanical voltage, shock wave, channel resistance, dielectric, pulse generator, differential equations."

A high accurate and convergent numerical framework for solving high-order nonlinear Volterra integro-differential equations

This article discuss about the application of Jacobi spectral and pseudo-spectral Galerkin techniques for solving high-order nonlinear Volterra integro-differential equations. We supply strict error analysis for Jacobi spectral and pseudo-spectral Galerkin methods(JSGM,JPSGM) under reasonable assumptions, which indicate that the approximate solution errors decay exponentially in the sense of L∞ and Lω2 norm. Some numerical experiments are presented to verify the theoretical results. Superior numerical results with respect to some recent methods are provided in tables and figures.