• Journal of applied mathematics & informatics
• /
• v.26 no.5_6
• /
• pp.1273-1287
• /
• 2008

In this paper, the numerical integration method for general singularly perturbed two point boundary value problems with mixed boundary conditions of both left and right end boundary layer is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Bulletin of the Korean Mathematical Society
• /
• v.27 no.2
• /
• pp.165-172
• /
• 1990

In this paper, we study a method which speeds up the convergence of monotone iterations and give some numerical examples of the types (2) and (3). In [2] this method has been applied to a system of hyperbolic differential equations with initial and boundary conditions.

• Advances in nano research
• /
• v.16 no.5
• /
• pp.473-487
• /
• 2024

The current study employs the nonlocal Timoshenko beam (NTB) theory and von-Kármán's geometric nonlinearity to develop a non-classic beam model for evaluating the nonlinear free vibration of bi-directional functionally-graded (BFG) nanobeams. In order to avoid the stretching-bending coupling in the equations of motion, the problem is formulated based on the physical middle surface. The governing equations of motion and the relevant boundary conditions have been determined using Hamilton's principle, followed by discretization using the differential quadrature method (DQM). To determine the frequencies of nonlinear vibrations in the BFG nanobeams, a direct iterative algorithm is used for solving the discretized underlying equations. The model verification is conducted by making a comparison between the obtained results and benchmark results reported in prior studies. In the present work, the effects of amplitude ratio, nanobeam length, material distribution, nonlocality, and boundary conditions are examined on the nonlinear frequency of BFG nanobeams through a parametric study. As a main result, it is observed that the nonlinear vibration frequencies are greater than the linear vibration frequencies for the same amplitude of the nonlinear oscillator. The study finds that the difference between the dimensionless linear frequency and the nonlinear frequency is smaller for CC nanobeams compared to SS nanobeams, particularly within the α range of 0 to 1.5, where the impact of geometric nonlinearity on CC nanobeams can be disregarded. Furthermore, the nonlinear frequency ratio exhibits an increasing trend as the parameter µ is incremented, with a diminishing dependency on nanobeam length (L). Additionally, it is established that as the nanobeam length increases, a critical point is reached at which a sharp rise in the nonlinear frequency ratio occurs, particularly within the nanobeam length range of 10 nm to 30 nm. These findings collectively contribute to a comprehensive understanding of the nonlinear vibration behavior of BFG nanobeams in relation to various parameters.

• Journal of applied mathematics & informatics
• /
• v.27 no.3_4
• /
• pp.933-941
• /
• 2009

In this paper, we consider p-Laplace equation which models the turbulent flow in a porous medium. Using a continuation principle (cf. [R. $Man{\acute{a}}sevich$ and J. Mawhin, Periodic solutions for nonlinear systems with p-Lplacian-like operators, J. Diff. Equa. 145(1998), 367-393]), we prove the existence of solutions for p-Laplace equation subject to periodic boundary conditions, under some sign and growth conditions for f. With the help of Leray-Schauder degree theory, the multiplicity of periodic solutions for p-Laplace equation is obtained under the similar conditions above and some known results are improved.

• Journal of applied mathematics & informatics
• /
• v.31 no.1_2
• /
• pp.221-228
• /
• 2013

The method of upper and lower solutions and the generalized quasilinearization technique is developed for the existence and approximation of solutions to boundary value problems for higher order fractional differential equations of the type $^c\mathcal{D}^qu(t)+f(t,u(t))=0$, $t{\in}(0,1),q{\in}(n-1,n],n{\geq}2$ $u^{\prime}(0)=0,u^{\prime\prime}(0)=0,{\ldots},u^{n-1}(0)=0,u(1)={\xi}u({\eta})$, where ${\xi},{\eta}{\in}(0,1)$, the nonlinear function f is assumed to be continuous and $^c\mathcal{D}^q$ is the fractional derivative in the sense of Caputo. Existence of solution is established via the upper and lower solutions method and approximation of solutions uses the generalized quasilinearization technique.

• Bulletin of the Korean Mathematical Society
• /
• v.51 no.2
• /
• pp.329-338
• /
• 2014

In this paper, we study the method of upper and lower solutions and develop the generalized quasilinearization technique for the existence and approximation of solutions to some three-point nonlocal boundary value problems associated with higher order fractional differential equations of the type $$^c{\mathcal{D}}^q_{0+}u(t)+f(t,u(t))=0,\;t{\in}(0,1)$$ $$u^{\prime}(0)={\gamma}u^{\prime}({\eta}),\;u^{\prime\prime}(0)=0,\;u^{\prime\prime\prime}(0)=0,{\ldots},u^{(n-1)}(0)=0,\;u(1)={\delta}u({\eta})$$, where, n-1 < q < n, $n({\geq}3){\in}\mathbb{N}$, 0 < ${\eta},{\gamma},{\delta}$ < 1 and $^c\mathcal{D}^q_{0+}$ is the Caputo fractional derivative of order q. The nonlinear function f is assumed to be continuous.

• Nonlinear Functional Analysis and Applications
• /
• v.27 no.2
• /
• pp.349-361
• /
• 2022

In this paper, we study the following nonlinear problem: $$\{-\Delta_{p}^{3}(x)u\;=\;{\lambda}V_{1}(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u\;=\;{\Delta}u\;{\Delta}^{2}u\;=\;0\;on\;{\partial}\Omega, $$ under adequate conditions on the exponent functions p, q and the weight function V₁. We prove the existence and nonexistence of eigenvalues for p(x)-triharmonic problem with Navier boundary value conditions on a bounded domain in ℝ^N. Our technique is based on variational approaches and the theory of variable exponent Lebesgue spaces.

• Structural Engineering and Mechanics
• /
• v.47 no.5
• /
• pp.599-622
• /
• 2013

Seismic response of two dimensional liquid tanks is numerically simulated using fully nonlinear velocity potential theory. Galerkin-weighted-residual based finite element method is used for solving the governing Laplace equation with fully nonlinear free surface boundary conditions and also for velocity recovery. Based on mixed Eulerian-Lagrangian (MEL) method, fourth order explicit Runge-Kutta scheme is used for time integration of free surface boundary conditions. A cubic-spline fitted regridding technique is used at every time step to eliminate possible numerical instabilities on account of Lagrangian node induced mesh distortion. An artificial surface damping term is used which mimics the viscosity induced damping and brings in numerical stability. Four earthquake motions have been suitably selected to study the effect of frequency content on the dynamic response of tank-liquid system. The nonlinear seismic response vis-a-vis linear response of rectangular liquid tank has been studied. The impulsive and convective components of hydrodynamic forces, e.g., base shear, overturning base moment and pressure distribution on tank-wall are quantified. It is observed that the convective response of tank-liquid system is very much sensitive to the frequency content of the ground motion. Such sensitivity is more pronounced in shallow tanks.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.20 no.4
• /
• pp.1251-1262
• /
• 1996

In this paper, a mathematical model for a belt driven system is proposed to analyse the vibration characteristics of the driving units with belts and the free and forced vibraiton anlyses are carried out. The mathematical model for a belt-driven system includes belts, pulleys, spindle and bearings. By using Hamilton's principle, four nonlinear governing equations and twelve nonlinear boundary conditions are derived. To linearize and discretize the nonlinear governing equations and boundary conditions, the perturbation method and Galerkin method are used. Also, the free vibration analyses for various parameters of a belt driven system, which are the tension of a belt, the length of a belt, the material properties of belts, the velocity of a velt and the mass of pulley are made. The forced vibration analyses of the system are performed and the dynamic responses for main parameters are anlysed with a belt driven system.

• Computers and Concrete
• /
• v.25 no.3
• /
• pp.255-266
• /
• 2020

This study presents comparation of fixed and viscos boundary condition effects on three-dimensional earthquake response and performance of a RCC dam considering linear and non-linear response. For this purpose, Cine RCC dam constructed in Aydın, Turkey, is selected in applications. The Drucker-Prager material model is considered for concrete and foundation rock in the nonlinear time-history analyses. Besides, hydrodynamic effect was considered in linear and non-linear dynamic analyses for both conditions. The hydrodynamic pressure of the reservoir water is modeled with the fluid finite elements based on the Lagrangian approach. The contact-target element pairs were used to model the dam-foundation-reservoir interaction system. The interface between dam and foundation is modeled with welded contact for both fixed and viscos boundary conditions. The displacements and principle stress components obtained from the linear and non-linear analyses are compared each other for empty and full reservoir cases. Seismic performance analyses considering demand-capacity ratio criteria were also performed for each case. According to numerical analyses, the total displacements and besides seismic performance of the dam increase by the effect of the viscous boundary conditions. Besides, hydrodynamic pressure obviously decreases the performance of the dam.