Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library

Noeiaghdam, Samad; Dreglea, Aliona; He, Ji‐Huan; Avazzadeh, Z.; Suleman, Muhammad; Araghi, Mohammad Ali Fariborzi; Sidorov, Denis; Sidorov, N. V.

doi:10.3390/sym12101730

Cited by 36 publications

(29 citation statements)

References 52 publications

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“…Viscoelasticity is one Symmetry 2021, 13, 256 2 of 17 of the fields of physics where the Volterra equations are often used [25]. The applications of Volterra equations in renewable energy is an example of their use in industry [30][31][32]. Volterra integro-differential equations are usually difficult to solve analytically; therefore, approximate solutions and numerical methods are often used [33].…”

Section: Introductionmentioning

confidence: 99%

Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

Минаков

Schick

2021

Symmetry

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An integro-differential equation describes the non-equilibrium thermal response of glass-forming substances with a dynamic (time-dependent) heat capacity to fast thermal perturbations. We found that this heat transfer problem could be solved analytically for a heat source with an arbitrary time dependence and different geometries. The method can be used to analyze the response to local thermal perturbations in glass-forming materials, as well as temperature fluctuations during subcritical crystal nucleation and decay. The results obtained can be useful for applications and a better understanding of the thermal properties of glass-forming materials, polymers, and nanocomposites.

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Section: Introductionmentioning

confidence: 99%

Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

Минаков

Schick

2021

Symmetry

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“…He [34][35][36] proposed a new perturbation technique coupled with the homotopy technique where the equation's small parameters were not required leading to eliminating the limitations of the traditional techniques. Noeiaghdam et al [37] proposed the HPM to study the second-kind linear Volterra integral equations with a discontinuous kernel. They validated the solution's accuracy by analyzing the convergence and error of the studied formulation.…”

Section: Introductionmentioning

confidence: 99%

Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Hamed

Kandil

2021

Symmetry

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Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

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“…Many years ago, the fractional differential equations were proven by researchers, showing that the fractional differential equations are a powerful tool in studying the problems of fractal geometry and fractal dynamics. Fractional differential equations furthermore show many advantages in modeling the important phenomena in many fields such as electromagnetic, fluid flow, acoustics, electrochemistry, and material science [7][8][9][10]. Is there a question in the financial market that the fractional differential equation can be applied?…”

Section: Introductionmentioning

confidence: 99%

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Ampun

Sawangtong

2021

Mathematics

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In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

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Error Estimation of the Homotopy Perturbation Method to Solve Second Kind Volterra Integral Equations with Piecewise Smooth Kernels: Application of the CADNA Library

Cited by 36 publications

References 52 publications

Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

Integro-Differential Equation for the Non-Equilibrium Thermal Response of Glass-Forming Materials: Analytical Solutions

Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

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