In the research paper, the authors exploit the definition of a new class of fractional integral operators, recently proposed by Jarad et al. (Adv. Differ. Equ. 2017:247, 2017), to define a new class of generalized k-fractional integral operators and develop a generalization of the reverse Minkowski inequality involving the newly introduced fractional integral operators. The two new theorems correlating with this inequality, including statements and verifications of other inequalities via the suggested k-fractional conformable integral operators, are presented.
Fundamental natural frequencies of rotating functionally graded cylindrical shells have been calculated through the improved wave propagation approach using three different volume fraction laws. The governing shell equations are obtained from Love’s shell approximations using improved rotating terms and the new equations are obtained in standard eigenvalue problem with wave propagation approach and volume fraction laws. The effects of circumferential wave number, rotating speed, length-to-radius, and thickness-to-radius ratios have been computed with various combinations of axial wave numbers and volume fraction law exponent on the fundamental natural frequencies of nonrotating and rotating functionally graded cylindrical shells using wave propagation approach and volume fraction laws with simply supported edge. In this work, variation of material properties of functionally graded materials is controlled by three volume fraction laws. This process creates a variation in the results of shell frequency. MATLAB programming has been used to determine shell frequencies for traveling mode (backward and forward) rotating motions. New estimations show that the rotating forward and backward simply supported fundamental natural frequencies increases with an increase in circumferential wave number, for Type I and Type II of functionally graded cylindrical shells. The presented results of backward and forward simply supported fundamental natural frequencies corresponding to Law I are higher than Laws II and III for Type I and reverse effects are found for Type II, depending on rotating speed. Our investigations show that the decreasing and increasing behaviors are noted for rotating simply supported fundamental natural frequencies with increasing length-to-radius and thickness-to-radius ratios, respectively. It is found that the fundamental frequencies of the forward waves decrease with the increase in the rotating speed, and the fundamental frequencies of the backward waves increase with the increase in the rotating speed. This investigation has been made with three different volume fraction laws of polynomial (Law I), exponential (Law II), and trigonometric (Law III). The presented numerical results of nonrotating isotropic and rotating functionally graded simply supported are in fair agreement with parts of other earlier numerical results.
This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the blacksolution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are blackevaluated in an iterative process with the help of homotopy perturbation method manipulating He’s polynomials. blackThe fractional derivatives are considered in the Caputo sense. The obtained results blackconfirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in blackthe form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with blackthe exact solution are presented.
Purpose
The main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system.
Design/methodology/approach
First, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution.
Findings
An example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system.
Originality/value
The work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work.
The He-Homotopy Perturbation Method and Laplace Transform, or HPLM, is a dependable combination that we take into consideration in this study in order to obtain the series solution of both linear and nonlinear higher order two-point boundary value problems. It is indicated that the corresponding technique generates better results in terms of accuracy, easy computation and is therefore recommended to be used for various physical models in engineering and science. Numerical results of five numerical examples are compared with the exact solutions as well as known analytical and numerical schemes to reveal the efficiency and excellence of the HPLM.
2010 Mathematics Subject Classification. Primary 26A33, 35A20; Secondary 33B15.
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