Nonlinear Integro-differential Equations Research Articles

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

ОБ ОЦЕНКЕ СНИЗУ РЕШЕНИЙ СЛАБО НЕЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ЗАПАЗДЫВАНИЯМИ

Sufficient conditions are established for lower bounds and tending to infinity as the argument of solutions to a weakly non-linear second-order Volterra integro-differential equation with delays on the half-axis grows unbounded. For this purpose, a non-standard method of reduction to a system, the method of transforming the equations of V. Volterra, the method of partial cutting, the method of integral inequalities with delays and the method of estimating solutions from below Yu.A. Ved are developed. It is shown that solutions with initial data from a certain manifold have lower bounds, and such solutions tend to infinity according to the exponential law with unlimited growth of the argument. This article proposes a technique for solving the problem of estimating from below on the half-line solutions of a weakly nonlinear Volterra-type integro-differential equation of the second order with delays, while the main role is played by the non-standard method of reduction to the system and the method of estimating solutions from below Yu.A.Ved (1965).

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in $$L^2$$ -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm.

Hasimoto transformation of general flows expressed in the Frenet frame

A one-dimensional space curve in R3 is a useful nonlinear medium for modeling vortex filaments and biological soft-matter capable of supporting a variety of wave motions. Hasimoto's transformation defines a mapping between the kinematic evolution of a space curve and nonlinear scalar equations evolving its intrinsic curve geometry. This mapping is quite robust and able to transform general vector fields expressed in the Frenet frame, resulting in a fully nonlinear integro-differential evolution equation, whose coefficient structure is defined by the coordinates of the flow in the Frenet frame. In this paper, we generalize the Hasimoto map to arbitrary flows defined on space curves, which we test against several existing kinematic flows. After this, we consider the time dynamics of length and bending energy to see that binormal flows are generally length preserving, and bending energy is fragile and unlikely to be conserved in the general case.

Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions

<abstract><p>This paper studies the solvability of boundary value problems for a nonlinear integro-differential equation. Converting the problem to an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE) is driven by using a suitable transformation. To investigate the existence and uniqueness of continuous solutions for the NVFIE under certain given conditions, the Krasnoselskii fixed point theorem and Banach contraction principle have been used. Finally, we numerically solve the NVFIE and study the rate of convergence using methods based on applying the modified Adomian decomposition method, and Liao's homotopy analysis method. As applications, some examples are provided to support our work.</p></abstract>

LONG-WAVE MODELS FOR A COUNTERCURRENT FLOW OF LIQUID FILM AND TURBULENT GAS FLOW

The paper considers the countercurrent flow of a vertically falling liquid film and a turbulent gas. In the case of small Reynolds numbers and when certain conditions for the characteristics of such a flow are met, the problem is reduced to one model equation for the film thickness. For long-wave weakly nonlinear disturbances, various nonlinear integrodifferential equations are obtained from this equation for the deviation of film thickness from the undisturbed level. Their specific type depends on the values of the magnitude of one parameter. This parameter is the ratio of the friction force from the side of the gas acting on the unperturbed film flow to the force of gravity. It is shown that, in addition to the countercurrent flow, such models also appear in a number of other gas-film flows. Both their radical differences and some common features are discussed. Several universal solutions of these model equations are presented.

Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

<abstract> <p>Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (<bold>NFIDEq</bold>) in dual Banach space C<sub>E</sub> (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (<bold>BMA</bold>) method as a numerical method. The major motivation for selecting the <bold>BMA</bold> approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the <bold>NFIDEq</bold> to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.</p> </abstract>

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Existence and data dependence results for neutral fractional order integro-differential equations

<abstract><p>We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.</p></abstract>

Stieltjes integral boundary value problem involving a nonlinear multi-term Caputo-type sequential fractional integro-differential equation

<abstract><p>In this article, we analyze the existence and uniqueness of mild solution to the Stieltjes integral boundary value problem involving a nonlinear multi-term, Caputo-type sequential fractional integro-differential equation. Krasnoselskii's fixed-point theorem and the Banach contraction principle are utilized to obtain the existence and uniqueness of the mild solution of the aforementioned problem. Furthermore, the Hyers-Ulam stability is obtained with the help of established methods. Our proposed model contains both the integer order and fractional order derivatives. As a result, the exponential function appears in the solution of the model, which is a fundamental and naturally important function for integer order differential equations and its many properties. Finally, two examples are provided to illustrate the key findings.</p></abstract>

Dynamic buckling of plate made of glass reinforced plastic under rapidly increasing shear load

The research object of this work is a clamped rectangular plate made of glass-reinforced plastic. The dynamic problem of stability of the plate under rapidly increasing shear load is considered. Within the Kirchhoff–Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation. By the Bubnov–Galerkin method, based on a polynomial approximation of the deflection, the problem was reduced to solving systems of nonlinear ordinary integro-differential equations. With a weakly singular Koltunov–Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. The plate’s dynamic behavior was investigated depending on the plate’s geometric and physic parameters. The importance of considering the viscoelastic properties of the material is shown.

Existence and stability results for impulsive $ (k, \psi) $-Hilfer fractional double integro-differential equation with mixed nonlocal conditions

<abstract><p>This paper investigates a class of nonlinear impulsive fractional integro-differential equations with mixed nonlocal boundary conditions (multi-point and multi-term) that involves $ (\rho_{k}, \psi_{k}) $-Hilfer fractional derivative. The main objective is to prove the existence and uniqueness of the solution for the considered problem by means of fixed point theory of Banach's and O'Regan's types, respectively. In this contribution, the transformation of the considered problem into an equivalent integral equation is necessary for our main results. Furthermore, the nonlinear functional analysis technique is used to investigate various types of Ulam's stability results. The applications of main results are guaranteed with three numerical examples.</p></abstract>

On some nonlinear fractional integro-differential equation and its connection with the Riemann-Liouville fractional integral in Lebesgue spaces

On some nonlinear fractional integro-differential equation and its connection with the Riemann-Liouville fractional integral in Lebesgue spaces

Existence, uniqueness and stability of solutions for generalized proportional fractional hybrid integro-differential equations with Dirichlet boundary conditions

<abstract><p>In this work, the existence of solutions for nonlinear hybrid fractional integro-differential equations involving generalized proportional fractional (GPF) derivative of Caputo-Liouville-type and multi-term of GPF integrals of Reimann-Liouville type with Dirichlet boundary conditions is investigated. The analysis is accomplished with the aid of the Dhage's fixed point theorem with three operators and the lower regularized incomplete gamma function. Further, the uniqueness of solutions and their Ulam-Hyers-Rassias stability to a special case of the suggested hybrid problem are discussed. For the sake of corroborating the obtained results, an illustrative example is presented.</p></abstract>

Convergence analysis for the fractional decomposition method applied to class of nonlinear fractional Fredholm integro-differential equation

For scientists conducting research, fractional integral differential equation analysis is crucial. Therefore, in this study, we investigate analysis utilizing a novel method called the fractional decomposition method, which is applicable to fractional nonlinear fractional Fredholm integro-differential equations. Then, we apply the approach to five test problems for a general fractional derivative [Formula: see text] involving fractional Fredholm integro-differential equations. To the best of our knowledge, we are the first to ever do so because of the very complicated calculations involved when dealing with the general case [Formula: see text]. For fractional Fredholm integral-differential equations, we provide both exact and approximate solutions. Throughout this work, the fractional Caputo derivative is discussed. This technique leads us to say that the method is precise, accurate, and efficient, according to the theoretical analysis.

AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR SECOND ORDER ORDINARY HYBRID INTEGRODIFFERENTIAL EQUATIONS

In this paper, we establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear second order ordinary hybrid integrodifferential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2022) and Dhage et al. (2022). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Finally, our main abstract results are also illustrated with a couple of numerical examples.

On the approximate boundary controllability of some partial functional integrodifferential equations with finite delay in Banach spaces

<p style='text-indent:20px;'>This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration.</p>

Dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings

The subject of this work is the study of local dynamics of full-coupled chains of a great number of oscillators with a large delay in couplings. From a discrete model describing the dynamics of a great number of coupled oscillators, a transition has been made to a nonlinear integro-differential equation, continuously depending on time and space variable. A class of full-coupled systems has been considered. The main assumption is that the amount of delay in the couplings is large enough. This assumption opens the way to the use of special asymptotic methods of study. The parameters under which the critical case is realized in the problem of the equilibrium state stability have been distinguished. It is shown that they have infinite dimension. The analogues of normal forms — nonlinear boundary value problems of Ginzburg–Landau type have been constructed. In some cases, these boundary value problems contain integral components too. Their nonlocal dynamics describes the behavior of all solutions of the original equations in the balance state neighbourhood. Methods. As applied to the considered problems, methods of constructing quasinormal forms on central manifolds are developed. An algorithm for constructing the asymptotics of solutions based on the use of quasinormal forms for determining slowly varying amplitudes has been created. Results. Quasinormal forms that determine the dynamics of the original boundary value problem have been constructed. The dominant terms of asymptotic approximations for solutions of the considered chains have been obtained. On the basis of the given statements, a number of interesting dynamical effects have been revealed. For example, an infinite alternation of direct and reverse bifurcations when the delay coefficient increases. Their distinguishing feature is that they have two or three spatial variables.

Novel algorithms to approximate the solution of nonlinear integro-differential equations of Volterra-Fredholm integro type

<abstract><p>This study is devoted to examine the existence and uniqueness behavior of a nonlinear integro-differential equation of Volterra-Fredholm integral type in continues space. Then, we examine its solution by modification of Adomian and homotopy analysis methods numerically. Initially, the proposed model is reformulated into an abstract space, and the existence and uniqueness of solution is constructed by employing Arzela-Ascoli and Krasnoselskii fixed point theorems. Furthermore, suitable conditions are developed to prove the proposed model's continues behavior which reflects the stable generation. At last, three test examples are presented to verify the established theoretical concepts.</p></abstract>

Nonlinear vibrations of earth structures

This article offers a detailed analysis of the current state of plane structure dynamics, addressing the complexities posed by non-homogeneous materials exhibiting nonlinear elastic and viscoelastic properties during real-life operations. The study proposes a comprehensive mathematical model and algorithm to investigate the dynamic behavior of such structures, employing the non-linear hereditary Boltzmann-Volterra theory to describe viscoelastic material properties accurately. Nonlinear oscillatory systems are analyzed using Lagrange equations based on the d’Alembert principle. The problem is approached through a multi-step process. Initially, the linear elastic problem of the structure’s natural oscillations is solved to determine its natural frequencies and modes of oscillations. Subsequently, these eigenmodes are employed as coordinate functions to address forced nonlinear oscillations in viscoelastic non-homogeneous systems. The complexity of the problem necessitates solving a Cauchy problem comprising a system of nonlinear integrodifferential equations. To illustrate the methodology, the study examines the Gissarak earth dam, considering real operational conditions and nonlinear, viscoelastic material properties near resonant modes of vibrations. Utilizing numerical methods, the dynamic behavior of the structure is analyzed, assessing displacements and stress components at different time points under non-stationary kinematic action. Stress concentration regions within the structure are identified for resonant vibrations, allowing the evaluation of its strength. Furthermore, the impact of nonlinear elasticity and viscoelasticity on the structural dynamics is quantified. This research provides valuable insights into the behavior of plane non-homogeneous structures, considering real-world scenarios and material complexities, ultimately contributing to an improved understanding of structural dynamics and facilitating the identification and mitigation of potential structural challenges.