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We propose the conjecture that for any positive integers r and n with n > 2, there do not exist 2r + 1 consecutive positive integers in natural order such that the sum of n-th powers of the first r + 1 integers equals the sum of n-th powers of the subsequent r integers, i.e., there are no positive integers r, m and n, where r < m and n > 2, satisfying (m – r)n + (m – r + 1)n + … + mn = (m + 1)n + (m + 2)n + … + (m + r)n. We prove that the conjecture is true for the cases n = 3 and n = 4. We also verified by using Mathematica that the conjecture is true for the cases 3 < n < 10 and m < 5000.

The fractional oscillation equation with two fractional derivative terms in the sense of Caputo, where the orders $$\alpha$$ α and $$\beta$$ β satisfy $$1 1 < α ≤ 2 and $$0 0 < β ≤ 1 , is investigated, and the unit step response and two initial value responses are obtained in two forms by using different methods of inverse Laplace transform. The first method yields series solutions with the nonnegative powers of t, which converge fast for small t. The second method is our emphasis, where the complex path integral formula of the inverse Laplace transform is used. In order to determine singularities of integrand we first seek for the roots of the characteristic equation, which is a transcendental equation with four parameters, two coefficients and two noninteger power exponents. The existence conditions and properties of the roots on the principal Riemann surface are given. Based on the results on the characteristic equation, we derive these responses as a sum of a classical exponentially damped oscillation, which vanishes in an indicated case, and an infinite integral of the Laplace type, which converges fast for large t, with a steady component in the unit step response. Asymptotic behaviors of solutions for large t are derived as algebraic decays in negative power laws characterized by the orders $$\alpha$$ α and $$\beta$$ β . The fractional system exhibits a magical transition from oscillation to monotonic decay in negative power law.

The modified decomposition method is applied to analyze the transverse vibrations of tapered rotating beams incorporating both axial centrifugal stiffening and flexible end constraints. Unlike prio...

The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 < β < 1) and $_{ - \infty }D_t^\alpha x$ (1 < α < 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 < β < 1) and α (1 < α < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.

Purpose The purpose of this study is to investigate viscoelastic properties for the constitutive equation in terms of distributed-order derivatives. Design/methodology/approach The authors considered the steady oscillatory shear flow between two parallel plates (one is fixed and another oscillates in its own plane), and then examined the effects of different forms of the order-weight functions. Findings The constitutive equation in terms of distributed-order derivatives can characterize viscoelastic properties. The order-weight functions can effectively describe viscoelasticity. Originality/value Model the viscoelastic constitutive equation in terms of distributed-order derivatives, where order-weight functions can select to respond viscoelastic properties.

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system.

The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator − ∞ D t β , ${}_{-\infty} D_t^\beta,$ where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.

We conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the order α of the fractional derivative and the parameter γ parameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameter γ, not exactly like the case of single fractional-order derivative for the order α. The case of the distributed-order derivative provides us more options for the weight function and parameters.

Purpose The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications. Design/methodology/approach The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids. Findings The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature. Originality/value This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

In this paper, a six-parameter fractional constitutive model was investigated by using the Laplace transform and its complex inverse formula. Our model includes all of the four types of viscoelasticity by the different values of two coefficients. We expressed the creep compliance and relaxation modulus as infinite real integrals which can be numerically calculated readily by the built-in numerical quadrature command of MATHEMATICA 11. Also, the infinite integration in the form of ∫ 0 + ∞ P ( r ) e − rt dr is convenient for numerical computation and theory analysis. Finally, the stress responses to harmonic strain were considered and the stress-strain hysteresis loops were displayed.

The response of a one-degree-of-freedom distributed-order vibration system to a harmonic driving was considered. For the steady case, the lower limit of integration in the definition of fractional derivatives was taken as negative infinity. The response amplitude, phase difference between driving and response, and the effects of distributed-order derivative on damping and resonance were investigated in detail. The results display that the parameter in distributed-order derivative can effectively characterize the viscoelasticity.

The steady-state response to periodic excitation in the linear fractional vibration system was considered by using the fractional derivative operator . First we investigated the response to the harmonic excitation in the form of complex exponential function. The amplitude-frequency relation and phase-frequency relation were derived. The effect of the fractional derivative term on the stiffness and damping was discussed. For the case of periodic excitation, we decompose the periodic excitation into a superposition of harmonic excitations by using the Fourier series, and then utilize the results for harmonic excitations and the principle of superposition, where our adopted tactics avoid appearing a fractional power of negative numbers to overcome the difficulty in fractional case. Finally we demonstrate the proposed method by three numerical examples.

The influence of the surface energy on the instability of nano-structures under the electrostatic force has been investigated in recent years by different researchers. It appears that in all prior research, the response of all structures becomes softer due to the surface effects. In the present study, the pull-in instability of a NEMS device incorporating the electrostatic force and Casimir intermolecular attraction for different values of the surface parameter is investigated by the Duan–Rach method of determined coefficients (MDC) in order to identify the remarkable effect of the surface energy. Although the obtained results verify the behavior of such structures in presence of the fringing field and the Casimir attraction same as the previous investigations, however the incremental effects of the surface energy cause the aforementioned structures to behave more stiffly in contrast.