This review explores multi-directional functionally graded(MDFG)nanostructures,focusing on their material characteristics,modeling approaches,and mechanical behavior.It starts by classifying different types of functio...This review explores multi-directional functionally graded(MDFG)nanostructures,focusing on their material characteristics,modeling approaches,and mechanical behavior.It starts by classifying different types of functionally graded(FG)materials such as conventional,axial,bi-directional,and tri-directional,and the material distribution models like power-law,exponential,trigonometric,polynomial functions,etc.It also discusses the application of advanced size-dependent theories like Eringen’s nonlocal elasticity,nonlocal strain gradient,modified couple stress,and consistent couple stress theories,which are essential to predict the behavior of structures at small scales.The review covers the mechanical analysis of MDFG nanostructures in nanobeams,nanopipes,nanoplates,and nanoshells and their dynamic and static responses under different loading conditions.The effect of multi-directional material gradation on stiffness,stability and vibration is discussed.Moreover,the review highlights the need for more advanced analytical,semi-analytical,and numerical methods to solve the complex vibration problems ofMDFG nanostructures.It is evident that the continued development of these methods is crucial for the design,optimization,and real-world application of MDFG nanostructures in advanced engineering fields like aerospace,biomedicine,and micro/nanoelectromechanical systems(MEMS/NEMS).This study is a reference for researchers and engineers working in the domain of MDFG nanostructures.展开更多
This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solu...This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method （VIM）. This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. （2008） 178 1917] along with homotopy perturbation method （HPM） and [Int. Commun. Heat Mass Transfer （2012） 39 30] in the special cases to demonstrate the validity and applicability.展开更多
This paper presents the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order.Homotopy Perturbation Method(HPM)is used to obtain...This paper presents the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order.Homotopy Perturbation Method(HPM)is used to obtain the dynamic response with respect to unit impulse load.Obtained results are depicted in term of plots.Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan(2007)to show the effectiveness and validation of the present method.展开更多
The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as ...The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.展开更多
The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties...The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 〈 α≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method（ADM） symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.展开更多
The present article investigates the effect of Coriolis constant on the solution of the Geophysical Korteweg-de Vries(gKdV)equation.As such,the Homotopy Perturbation Method(HPM)has been applied here for solving the no...The present article investigates the effect of Coriolis constant on the solution of the Geophysical Korteweg-de Vries(gKdV)equation.As such,the Homotopy Perturbation Method(HPM)has been applied here for solving the nonlinear gKdV equation.Present results are compared with existing results that are available in the literature and they are found to be in good agreement.Then the Coriolis term has been considered in terms of the interval to form interval Geophysical Korteweg-de Vries(IgKdV)equation.IgKdV equation has been solved by HPM to analyse the effect of Coriolis.From this analysis,it has been concluded that the constant of Coriolis is directly proportional to wave height and inversely proportional to wavelength.The presence of the Coriolis term in gKdV equation has a remarkable change in the shape of the solution.展开更多
文摘This review explores multi-directional functionally graded(MDFG)nanostructures,focusing on their material characteristics,modeling approaches,and mechanical behavior.It starts by classifying different types of functionally graded(FG)materials such as conventional,axial,bi-directional,and tri-directional,and the material distribution models like power-law,exponential,trigonometric,polynomial functions,etc.It also discusses the application of advanced size-dependent theories like Eringen’s nonlocal elasticity,nonlocal strain gradient,modified couple stress,and consistent couple stress theories,which are essential to predict the behavior of structures at small scales.The review covers the mechanical analysis of MDFG nanostructures in nanobeams,nanopipes,nanoplates,and nanoshells and their dynamic and static responses under different loading conditions.The effect of multi-directional material gradation on stiffness,stability and vibration is discussed.Moreover,the review highlights the need for more advanced analytical,semi-analytical,and numerical methods to solve the complex vibration problems ofMDFG nanostructures.It is evident that the continued development of these methods is crucial for the design,optimization,and real-world application of MDFG nanostructures in advanced engineering fields like aerospace,biomedicine,and micro/nanoelectromechanical systems(MEMS/NEMS).This study is a reference for researchers and engineers working in the domain of MDFG nanostructures.
基金the UGC, Government of India, for financial support under the Rajiv Gandhi National Fellowship (RGNF)
文摘This paper investigates the numerical solution of the uncertain inverse heat conduction problem. Uncertainties present in the system parameters are modelled through triangular convex normalized fuzzy sets. In the solution process, double parametric forms of fuzzy numbers are used with the variational iteration method （VIM）. This problem first computes the uncertain temperature distribution in the domain. Next, when the uncertain temperature measurements in the domain are known, the functions describing the uncertain temperature and heat flux on the boundary are reconstructed. Related example problems are solved using the present procedure. We have also compared the present results with those in [Inf. Sci. （2008） 178 1917] along with homotopy perturbation method （HPM） and [Int. Commun. Heat Mass Transfer （2012） 39 30] in the special cases to demonstrate the validity and applicability.
文摘This paper presents the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order.Homotopy Perturbation Method(HPM)is used to obtain the dynamic response with respect to unit impulse load.Obtained results are depicted in term of plots.Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan(2007)to show the effectiveness and validation of the present method.
文摘The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.
基金the UGC,Government of India,for financial support under Rajiv Gandhi National Fellowship(RGNF)
文摘The fractional diffusion equation is one of the most important partial differential equations（PDEs） to model problems in mathematical physics. These PDEs are more practical when those are combined with uncertainties. Accordingly, this paper investigates the numerical solution of a non-probabilistic viz. fuzzy fractional-order diffusion equation subjected to various external forces. A fuzzy diffusion equation having fractional order 0 〈 α≤ 1 with fuzzy initial condition is taken into consideration. Fuzziness appearing in the initial conditions is modelled through convex normalized triangular and Gaussian fuzzy numbers. A new computational technique is proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy diffusion equation is converted first into an interval-based fuzzy differential equation. Next, this equation is transformed into crisp form by using the proposed double parametric form of fuzzy numbers. Finally, the same is solved by Adomian decomposition method（ADM） symbolically to obtain the uncertain bounds of the solution. Computed results are depicted in terms of plots. Results obtained by the proposed method are compared with the existing results in special cases.
基金The authors are thankful to Board of Research in Nu-clear Sciences(BRNS),Mumbai,India(Grant Number:36(4)/40/46/2014-BRNS)for the support and funding to carry out the present research work.
文摘The present article investigates the effect of Coriolis constant on the solution of the Geophysical Korteweg-de Vries(gKdV)equation.As such,the Homotopy Perturbation Method(HPM)has been applied here for solving the nonlinear gKdV equation.Present results are compared with existing results that are available in the literature and they are found to be in good agreement.Then the Coriolis term has been considered in terms of the interval to form interval Geophysical Korteweg-de Vries(IgKdV)equation.IgKdV equation has been solved by HPM to analyse the effect of Coriolis.From this analysis,it has been concluded that the constant of Coriolis is directly proportional to wave height and inversely proportional to wavelength.The presence of the Coriolis term in gKdV equation has a remarkable change in the shape of the solution.
