• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.8 no.2
• /
• pp.23-38
• /
• 2004

We consider finite element methods applied to a class of quasi parabolic integro-differential equations in $R^d$. Global strong superconvergence, which only requires that partitions are quasi-uniform, is investigated for the error between the approximate solution and the Sobolev-Volterra projection of the exact solution. Two order superconvergence results are demonstrated in $W^{1,p}(\Omega)\;and\;L_p(\Omega)$, for $2\;{\leq}p\;<\;{\infty}$.

• Korean Journal of Mathematics
• /
• v.14 no.2
• /
• pp.185-202
• /
• 2006

In this paper, using the Black-Scholes analysis, we will derive the partial differential equation of convertible bonds with both non-stochastic and stochastic interest rate. We also find numerical solutions of convertible bonds equation with known interest rate using the finite element method.

• International Journal of Highway Engineering
• /
• v.17 no.5
• /
• pp.1-10
• /
• 2015

PURPOSES : A finite difference model considering snow melting process on porous asphalt pavement was derived on the basis of heat transfer and mass transfer theories. The derived model can be applied to predict the region where black-ice develops, as well as to predict temperature profile of pavement systems where a de-icing system is installed. In addition, the model can be used to determined the minimum energy required to melt the ice formed on the pavement. METHODS : The snow on the porous asphalt pavement, whose porosity must be considered in thermal analysis, is divided into several layers such as dry snow layer, saturated snow layer, water+pavement surface, pavement surface, and sublayer. The mass balance and heat balance equations are derived to describe conductive, convective, radiative, and latent transfer of heat and mass in each layer. The finite differential method is used to implement the derived equations, boundary conditions, and the testing method to determine the thermal properties are suggested for each layer. RESULTS: The finite differential equations that describe the icing and deicing on pavements are derived, and we have presented them in our work. The framework to develop a temperature-forecasting model is successfully created. CONCLUSIONS : We conclude by successfully creating framework for the finite difference model based on the heat and mass transfer theories. To complete implementation, laboratory tests required to be performed.

• Journal of applied mathematics & informatics
• /
• v.39 no.5_6
• /
• pp.623-641
• /
• 2021

In this paper, numerical treatment of singularly perturbed parabolic delay differential equations is considered. The considered problem have small delay on the spatial variable of the reaction term. To treat the delay term, Taylor series approximation is applied. The resulting singularly perturbed parabolic PDEs is solved using Crank Nicolson method in temporal direction with non-standard finite difference method in spatial direction. A detail stability and convergence analysis of the scheme is given. We proved the uniform convergence of the scheme with order of convergence O(N^-1 + (∆t)²), where N is the number of mesh points in spatial discretization and ∆t is mesh length in temporal discretization. Two test examples are used to validate the theoretical results of the scheme.

• Coupled systems mechanics
• /
• v.1 no.4
• /
• pp.345-360
• /
• 2012

In this article we have presented a unique representation for interval arithmetic. The traditional interval arithmetic is transformed into crisp by symbolic parameterization. Then the proposed interval arithmetic is extended for fuzzy numbers and this fuzzy arithmetic is used as a tool for uncertain finite element method. In general, the fuzzy finite element converts the governing differential equations into fuzzy algebraic equations. Fuzzy algebraic equations either give a fuzzy eigenvalue problem or a fuzzy system of linear equations. The proposed methods have been used to solve a test problem namely heat conduction problem along with fuzzy finite element method to see the efficacy and powerfulness of the methodology. As such a coupled set of fuzzy linear equations are obtained. These coupled fuzzy linear equations have been solved by two techniques such as by fuzzy iteration method and fuzzy eigenvalue method. Obtained results are compared and it has seen that the proposed methods are reliable and may be applicable to other heat conduction problems too.

• Structural Engineering and Mechanics
• /
• v.57 no.5
• /
• pp.861-876
• /
• 2016

The determination of the critical buckling load of multistory structures is important since this load is used in second order analysis. It is more realistic to determine the critical buckling load of multistory structures using the whole system instead of independent elements. In this study, a method is proposed for designating the system critical buckling load of torsion-free structures of which the load-bearing system consists of frames and shear walls. In the method presented, the multistory structure is modeled in accordance with the continuous system calculation model and the differential equation governing the stability case is solved using the differential transform method (DTM). At the end of the study, an example problem is solved to show the conformity of the presented method with the finite elements method (FEM).

• Structural Engineering and Mechanics
• /
• v.77 no.1
• /
• pp.37-46
• /
• 2021

In this study, a method for free vibration analysis of wall-frame systems built on weak soil is proposed. In the development of the method, the wall-frame system that constitutes the superstructure was modeled as flexural-shear beam. In the study, it is accepted that the soil layers are isotropic, homogeneous and elastic, and the waves are only vertical propagating shear waves. Based on this assumption, the soil layer below is modeled as an equivalent shear beam. Then the differential equation system that represented the behavior of the whole system was written for both regions in a separate way. Natural periods were obtained by solving the differential equations by employing boundary conditions. At the end of the study, two examples were solved and the suitability of the proposed method to the Finite Element Method was evaluated.

• Journal of applied mathematics & informatics
• /
• v.31 no.1_2
• /
• pp.299-309
• /
• 2013

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The fractional derivatives are presented in terms of Caputo sense. The application of the method to fractional BVPs leads to algebraic systems which can be solved by an appropriate method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

• Structural Engineering and Mechanics
• /
• v.80 no.5
• /
• pp.491-501
• /
• 2021

A difference in subgrade settlement between two rails of a track manifests as lateral differential subgrade settlement. This settlement causes unsteadiness in the motion of trains passing through the corresponding area. To illustrate the effect of lateral differential subgrade settlement on the dynamic response of a vehicle-track coupling system, a three-dimensional vehicle-track-subgrade coupling model was formulated by combining the vehicle-track dynamics theory and the finite element method. The wheel/rail force, car body acceleration, and derailment factor are chosen as evaluation indices of the system dynamic response. The effects of the amplitude and wavelength of lateral differential subgrade settlement as well as the driving speed of the vehicle are analyzed. The study reveals the following: The dynamic responses of the vehicle-track system generally increase linearly with the driving speed when the train passes through a lateral subgrade settlement area. The wheel/rail force acting on a rail with a large settlement exceeds that on a rail with a small settlement. The dynamic responses of the vehicle-track system increase with the amplitude of the lateral differential subgrade settlement. For a 250-km/h train speed, the proposed maximum amplitude for a lateral differential settlement with a wavelength of 20 m is 10 mm. The dynamic responses of the vehicle-track system decrease with an increase in the wavelength of the lateral differential subgrade settlement. To achieve a good operation quality of a train at a 250-km/h driving speed, the wavelength of a lateral differential subgrade settlement with an amplitude of 20 mm should not be less than 15 m. Monitoring lateral differential settlements should be given more emphasis in routine high-speed railway maintenance and repairs.

• Water for future
• /
• v.22 no.2
• /
• pp.233-239
• /
• 1989

The partial differential equation of the groundwater flow was reduced to an ordinary differential equation by the Boltzmann transformation. Its numerical solutions were obtained by the finite difference method and the new method to get the initial missing slope using the Richardson method and the finite difference equation was proposed. The solutions computed by the newly proposed method were compared with investigator's computations and they showed a satisfactory agreement and that the proposed method is easy and simple to get solutions.