• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제15권3호
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• pp.201-222
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• 2011

In the present work, the mathematical model of wire coating in a straight annular die is developed for unsteady second grade fluid in the form of partial differential equation. The Optimal Homotopy Asymptotic Method (OHAM) is applied for obtaining the solution of the model problem. This method provides us a suitable way to control the convergence of the series solution using the auxiliary constants which are optimally determined.

• Structural Engineering and Mechanics
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• 제50권4호
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• pp.487-500
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• 2014

An analytical solution for the nonlinear in-plane free oscillations of the suspended cable which contains the quadratic and cubic nonlinearities is investigated via the homotopy analysis method (HAM). Different from the existing analytical technique, the HAM is indeed independent of the small parameter assumption in the nonlinear vibration equation. The nonlinear equation is established by using the extended Hamilton's principle, which takes into account the effects of the geometric nonlinearity and quasi-static stretching. A non-zero equilibrium position term is introduced due to the quadratic nonlinearity in order to guarantee the rule of the solution expression. Therefore, the mth-order analytic solutions of the corresponding equation are explicitly obtained via the HAM. Numerical results show that the approximate solutions obtained by using the HAM are in good agreement with the numerical integrations (i.e., Runge-Kutta method). Moreover, the HAM provides a simple way to adjust and control the convergent regions of the series solutions by means of an auxiliary parameter. Finally, the effects of initial conditions on the linear and nonlinear frequency ratio are investigated.

• Biomaterials and Biomechanics in Bioengineering
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• 제5권1호
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• pp.51-64
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• 2020

In this study, the mathematical model that describes blood cell development in the bone marrow (i.e., hematopoiesis) has been studied via the Homotopy Perturbation Method (HPM). The results from the present work compared very well with the numerical solutions from published literature. This work has shown that the HPM is viable for solving delay differential equations born from hematopoiesis problem. The influence of the proliferating cells loss rate, time delay rate and the phase re-entry rate on the population densities of both the proliferating and resting cells were also determined through the underlined procedure.

• 한국항공우주학회지
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• 제35권7호
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• pp.647-654
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• 2007

본 논문에서는 일정한 추력을 이용한 지구-달의 연료 최적 궤적해를 이용하여, 가변 추력기를 이용한 지구 탈출 궤적에서의 에너지 및 연료 최적 궤적을 설계하였다. 에너지 최적의 지구 탈출 궤적은 여러 차례 지구를 공전하게 되고, 궤도 천이 시간이 일반적인 궤도 천이 시간 보다 상대적으로 오래 소요되므로 최적화 문제를 해를 구하기가 쉽지 않다. 따라서 에너지 최적 해를 구하기 위하여, 초기 상태변수를 조정하면서 Discrete continuation의 기법을 적용하였다. 최적화 문제의 종말 조건은 일정한 추력기를 이용한 지구 탈출 궤적의 종말 조건을 이용하였고, 구속 시간은 일정한 추력기를 이용한 궤도 천이의 경우보다 큰 값으로 설정하였다. 한편, 연료 최적 궤적은 제어 입력의 형태가 불연속적이기 때문에 해를 구하기가 쉽지 않다. 따라서 연료 최적 궤적은 에너지 최적의 해와 호모토피(Homotopy) 기법을 적용시켜 그 해를 구하였다.

• Structural Engineering and Mechanics
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• 제61권2호
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• pp.209-220
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• 2017

The purpose of this paper is to study the geometrically nonlinear free vibration of functionally graded nano/micro beams (FGNBs) based on the modified couple stress theory. For practical applications, some analytical expressions of nonlinear frequencies for FGNBs on a nonlinear Pasternak foundation are developed. Hamilton's principle is employed to obtain nonlinear governing differential equations in the context of both Euler-Bernoulli and Timoshenko beam theories for a comprehensive investigation. The modified continuum theory contains one material length scale parameter to capture the size effect. The variation of two-constituent material along the thickness is modeled using Reddy's power-law. Also, the Mori-Tanaka method as an accurate homogenization technique is implemented to estimate the effective material properties of the FGNBs. The results are presented for both hinged-hinged and clamped-clamped boundary conditions. The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method and then the powerful method of homotopy analysis is utilized to obtain the semi-analytical solutions. Eventually, the presented analytical expressions are used to examine the influences of the length scale parameter, material gradient index, and elastic foundation on the nonlinear free vibration of FGNBs.

• Structural Engineering and Mechanics
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• 제68권1호
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• pp.103-119
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• 2018

In the present research, an attempt is made to obtain a semi analytical solution for both nonlinear natural frequency and forced vibration of embedded functionally graded double layered nanoplates with all edges simply supported based on nonlocal strain gradient elasticity theory. The interaction of van der Waals forces between adjacent layers is included. For modeling surrounding elastic medium, the nonlinear Winkler-Pasternak foundation model is employed. The governing partial differential equations have been derived based on the Mindlin plate theory utilizing the von Karman strain-displacement relations. Subsequently, using the Galerkin method, the governing equations sets are reduced to nonlinear ordinary differential equations. The semi analytical solution of the nonlinear natural frequencies using the homotopy analysis method and the exact solution of the nonlinear forced vibration through the Harmonic Balance method are then established. The results show that the length scale parameters give nonlinearity of the hardening type in frequency response curve and the increase in material length scale parameter causes to increase in maximum response amplitude, whereas the increase in nonlocal parameter causes to decrease in maximum response amplitude. Increasing the material length scale parameter increases the width of unstable region in the frequency response curve.

• Advances in concrete construction
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• 제15권5호
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• pp.303-312
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• 2023

The present study investigates the effects of Cattaneo-Christov thermal effects of stagnation point in Walters-B nanofluid flow through lubrication of power-law fluid by taking the slip at the interfacial condition. For the solution, the governing partial differential equation is transformed into a series of non-linear ordinary differential equations. With the help of hybrid homotopy analysis method; that consists of both the homotopy analysis and shooting method these equations can be solved. The influence of different involved constraints on quantities of interest are sketched and discussed. The viscoelastic parameter, slip parameters on velocity component and temperature are analyzed. The velocity varies by increase in viscoelastic parameter in the presence of slip parameter. The slip on the surface has major effect and mask the effect of stagnation point for whole slip condition and throughout the surface velocity remained same. Matched the present solution with previously published data and observed good agreement. It can be seen that the slip effects dominates the effects of free stream and for the large values of viscoelastic parameter the temperature as well as the concentration profile both decreases.

• Structural Engineering and Mechanics
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• 제5권5호
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• pp.577-586
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• 1997

In nonlinear finite element stability analysis of structures, the foremost necessary procedure is the computation to precisely locate a singular equilibrium point, at which the instability occurs. The present study describes global and local procedures for the computation of stability points including bifurcation points and limit points. The starting point, at which the procedure will be initiated, may be close to or arbitrarily far away from the target point. It may also be an equilibrium point or non-equilibrium point. Apart from the usual equilibrium path, bypass and homotopy path are proposed as the global path to the stability point. A local iterative method is necessary, when it is inspected that the computed path point is sufficiently close to the stability point.

• Advances in concrete construction
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• 제14권2호
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• pp.103-113
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• 2022

On the basis of fabrication, the utilization of nano material in numerous industrial and technological system, obtained the utmost significance in current decade. Therefore, the current investigation presents a theoretical disposition regarding the flow of electric conducting Williamson nanoliquid over a stretchable surface in the presence of the motile microorganism. The impact of thermal radiation and magnetic parameter are incorporated in the energy equation. The concentration field is modified by adding the influence of chemical reaction. Moreover, the splendid features of nanofluid are displayed by utilizing the thermophoresis and Brownian motion aspects. Compatible similarity transformation is imposed on the equations governing the problem to derive the dimensionless ordinary differential equations. The Homotopy analysis method has been implemented to find the analytic solution of the obtained differential equations. The implications of specific parameters on profiles of velocity, temperature, concentration and motile microorganism density are investigated graphically. Moreover, coefficient of skin friction, Nusselt number, Sherwood number and density of motile number are clarified in tabular forms. It is revealed that thermal radiation, thermophoresis and Brownian motion parameters are very effective for improvement of heat transfer. The reported investigation can be used in improving the heat transfer appliances and systems of solar energy.

• Advances in nano research
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• 제16권4호
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• pp.427-434
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• 2024

Latest advancement in field of fluid dynamics has taken nanofluid under consideration which shows large thermal conductance and enlarges property of heat transformation in fluids. Motivated by this, the key aim of the current investigation scrutinizes the influence of thermal radiation and magnetohydrodynamic on the laminar flow of an incompressible two-dimensional Williamson nanofluid over an inclined surface in the presence of motile microorganism. In addition, the impact of heat absorption/generation and Arrhenius activation energy is also examined. A mathematical modeled is developed which stimulate the physical flow problem. By using the compatible similarities, we transfer the governing PDEs into ODEs. The analytic approach based on Homotopy analysis method is introduced to impose the analytic solution by using Mathematica software. The impacts of distinct pertinent variable on velocity profiles are investigated through graphs.