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Numerical Methods for PDEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Hence, the first derivative at every point in a domain may be computed by taking a Fourier transform, multiplying by −iω, and then taking an inverse Fourier transform. By discretizing equations (224.5) and (224.6), the Fourier transforms can be performed by “fast Fourier transforms” (FFTs). The FFT is a fast numerical technique for determining the finite Fourier transform of a function that is defined on a set of equally spaced grid points.
Effect of nonlocality in the vibration of a microscale ribbon due to laser pulse
Published in Waves in Random and Complex Media, 2022
Abhik Sur, Sudip Mondal, M. Kanoria
Employing inversion of the Laplace transform to Equation (40), we have where α being roots of the equation and β being roots of the equation Employing the inversion of Finite Fourier transform, from (40), the flexural deflection of the ribbon is obtained by Using Equation (38), the temperature increment in Laplace transform domain is given by Moreover, on applying the inversion of the Laplace transform, we arrive at Applying the inversion of the finite Fourier transform, the temperature increment within the microribbon is given by The solutions corresponding to the case when the CF non-local parameter agree with the results of the existing literature [28] up to quite satisfactory level.
Transient temperature distribution in a rotating cylinder subject to a surface heat source and convective cooling
Published in Numerical Heat Transfer, Part A: Applications, 2022
M. Arizmendi, F. Veiga, A. Jiménez, A. Gil Del Val
Using Eq. (7), the finite Fourier transform of Eq. (6) with respect to the axial variable yields to an initial value problem defined by the following first-order ordinary differential equation: where the transformed initial condition () can be expressed for the solid cylinder and the hollow cylinder as: and
Effect on dynamic stress distribution by the shape of cavity in continuous inhomogeneous medium under SH waves incidence
Published in Mechanics of Advanced Materials and Structures, 2021
Zailin Yang, Guanxixi Jiang, Yunqiu Song, Yong Yang, Menghan Sun
Functionally graded materials (FGMs) are regarded as a series of special composite materials since the material parameters varies along a certain direction [7]. Due to the requirement of mechanical components with different special functions in practical engineering, wave propagation in functionally graded cylinders, shells and beams attracted lots of attention in many fields. Dorduncu et al. researched wave propagation in a functionally graded circular cylinder subjected to dynamic loads [8]. The material properties vary through the thickness direction with a power-law distribution. Then, the displacement and stress distribution in the cylinder was calculated with the aids of the Mori-Tanaka homogenization scheme and the finite difference method (FDM). Shen et al. studied the cutoff frequencies of the circumferential SH waves in cylinder shells [9]. The shells were made of functionally graded piezoelectric-piezomagnetic material. Utilizing the Wentzel-Kramers-Brillouin (WKB) method, the differential equations with variable coefficients was solved. Considering the application of soft materials, Wu et al. investigated torsional waves and longitudinal waves propagating in the functionally graded hollow cylinder [10]. For the purpose of providing theoretical guidance for NDT and designing tunable waveguides, dispersion relations for the two types of axisymmetric guided waves are derived analytically, and then influences of several parameters on wave propagation characteristics were discussed through numerical examples. SH wave scattering model of a functionally graded tube embedded in an infinite elastic medium was proposed by Kara and Aydogdu [11]. Finite Fourier Transform was applied to solve the governing equation in the tube and the infinite medium. The sear modulus of the tube varied linearly from inner surface to outer surface. Comparing the degraded results with Pao and Mow [12], the validity of the calculation was proved. Finally, the hoop stress and the radial stress at the outer interface of the tube were calculated and discussed.