Nonlinear Integro-differential Equations Research Articles

An Iterative Scheme for Solving Arbitrary-Order Nonlinear Volterra Integro-Differential Equations Involving Delay

This paper introduces an iterative-based numerical scheme for solving nonlinear fractional-order Volterra integro-differential equations involving delay. Additionally, we provide sufficient conditions for the existence and uniqueness of the solution. The composite trapezoidal rule is applied to approximate the integral involved in the equation, followed by discretizing the Caputo fractional derivative operator of arbitrary order $$\alpha \in (0,1)$$ by using the classical L1 scheme. Further, the Daftardar-Gejji and Jafari method is employed to solve the implicit algebraic equation. The convergence analysis and error bounds of the proposed scheme are presented. It is shown that the approximate solution converges to the exact solution with order $$( 2-\alpha ).$$ We illustrate the efficacy and applicability of the proposed method through a couple of examples.

Uniqueness of a nonlinear integro-differential equation with nonlocal boundary condition and variable coefficients

This paper studies the uniqueness of solutions to a two-term nonlinear fractional integro-differential equation with nonlocal boundary condition and variable coefficients based on the Mittag-Leffler function, Babenko’s approach, and Banach’s contractive principle. An example is also provided to illustrate the applications of our theorem.

Projection Methods for Neural Field Equations

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields and derive a priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice depends on the choice of the projector. Using an elementary time stepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector and one from the time discretization. We give examples of concrete projection methods: two collocation schemes (piecewise linear and spectral collocation) and two Galerkin schemes (finite element and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.

A Nonlinear Schrеdinger Equation for Gravity-Capillary Waves on Deep Water with Constant Vorticit

The surface gravity-capillary waves on deep water with constant vorticity in the regionbounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformal variables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained. bounded by the free surface and the infinitely deep plane bottom are considered. A nonlinear Schrödinger equation is derived from a system of exact nonlinear integro-differential equations in conformalvariables written in the implicit form taking into account surface tension. In deriving the nonlinear Schrödinger equation, the role of the mean flow is taken into account. The nonlinear Schrödinger equation is investigated for modulation instability. A soliton solution of the nonlinear Schrödinger equation that represents a soliton of the “ninth wave” type is obtained.

Analytical-based exact-kernel vibration and long-term creep stress and large deformation redistributions of the suddenly pressurized incompressible visco-hyperelastic thick cylinders

The long-term creep and short-term damped dynamic redistributions of the large deformations and stresses are investigated here for long, suddenly pressurized, rubber-like visco-hyperelastic thick-walled cylinders, for the first time. The extension of the Mooney–Rivlin strain energy density function to incompressible visco-hyperelastic materials is accomplished by using a hereditary integral based on the Prony series and an exponential relaxation kernel. To extract a more accurate/efficient formulation, the relaxation kernel is processed instead of the common approximate discretization of the time derivative of the stress tensor in the convolution integral. The resulting nonlinear integrodifferential equations whose number of terms grows with time are solved by a novel analytical approach in conjunction with the second-order Runge–Kutta time-marching, bound-varying trapezoidal technique, and an iterative/updating scheme. In comparison to the commercial finite element analysis codes, the present analytical formulation and solution algorithm lead to higher accuracies from the mathematical point of view. The visco-hyperelastic properties are extracted from previously reported experimental results, using an error minimization fitting technique. Results reveal that in contrast to the hyperelastic vessel, the oscillations of the displacements and stresses of the visco-hyperelastic cylinder occur around levels that grow with time. Moreover, smaller relaxation parameters lead to smaller displacements and stresses and the responses intend earlier to the steady-state condition for higher values of the stiffness elements of the Prony series. In long term, the mean stresses and mean radial displacements of the inner and outer layers of the cylinder grow asymptotically with time but the largest overshoots and amplitudes of the responses occur at the early times. However, smaller values of the relaxation parameter expedite the convergence to the steady-state condition.

Asymptotic stability of nonlinear neutral delay integro-differential equations

Abstract In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors.

The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator

Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations

In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach.

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.