An Analytical Survey on the Solutions of the Generalized Double-Order φ -Integrodifferential Equation

We study the existence of solutions for a newly configured model of a double-order integrodifferential equation including φ -Caputo double-order φ -integral boundary conditions. In this way, we use the Krasnoselskii and Leray-Schauder fixed point results. Also, we invoke the Banach contraction principle to confirm the uniqueness of the existing solutions. Finally, we provide three examples to illustrate our analytical findings.

Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative

<abstract><p>This investigation communicates with an initial value problem (IVP) of Hilfer-generalized proportional fractional ($ \mathcal{GPF} $) differential equations in the fuzzy framework is deliberated. By means of the Hilfer-$ \mathcal{GPF} $ operator, we employ the methodology of successive approximation under the generalized Lipschitz condition. Based on the proposed derivative, the fractional Volterra-Fredholm integrodifferential equations $ (\mathcal{FVFIE}s) $ via generalized fuzzy Hilfer-$ \mathcal{GPF} $ Hukuhara differentiability ($ \mathcal{HD} $) having fuzzy initial conditions are investigated. Moreover, the existence of the solution is proposed by employing the fixed-point formulation. The uniqueness of the solution is verified. Furthermore, we derived the equivalent form of fuzzy $ \mathcal{FVFIE}s $ which is supposed to demonstrate the convergence of this group of equations. Two appropriate examples are presented for illustrative purposes.</p></abstract>

On existence and uniqueness results for iterative mixed integrodifferential equation of fractional order

The objective of this work is to study the local existence, uniqueness, stability and other properties of solutions of iterative mixed integrodifferential equations of fractional order. The Successive Approximation Method is applied for the numerical solution of iterative mixed integrodifferential equations of fractional order.

Discrete-continuous simulation and analysis of forced oscillations of (heavy) rope of crane loading mechanism in its process

The analysis of forced oscillations and waveformations of heavy ropes of hoisting mechanisms of cranes in the processes of starting the latter is carried out. The main parameters of the resulting waveforms in the ropes for different ways of lifting the load ("with a pickup", "by weight") are determined. The influence of gravity on the rope and the longitudinal oscillations arising in it, as well as on the longitudinal waveforms arising inside it are investigated. The rope of the lifting mechanism, in this case, is considered as a tensile rod. The laws of motion of cross-sections of ropes are established, according to which the forces arising in ropes are minimized during the startup of hoisting mechanisms of cranes. The application of the integrodifferential equation, which adequately describes the oscillations of the rope system of cargo cranes, and also takes into account the frequencyindependent internal friction in these systems, which accompanies the oscillatory processes in the ropes during transient reagents for their differentiation by different ways of lifting loads. The parameters of the established harmonic oscillations of the rope system of cranes for long-term lifting of the load and taking into account the inertial properties of the rope (considerable length), as well as the magnitude of the static movement of the rope (for different ways of lifting loads) are determined.