• 호남수학학술지
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• 제30권4호
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• pp.631-635
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• 2008

In this paper we discuss on the formal linearization and exact solution of Klein-Gordon's equation (1) $u_{tt}-au_{xx}+bu-cu^3=0 a,b,c{\in}R^+$ So that we know an efficient method for constructing of particular solutions of some nonlinear partial differential equations is introduced.

• 대한수학회보
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• 제53권5호
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• pp.1411-1425
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• 2016

We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brownian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated, and then an expression of the value for European options is obtained using the fundamental solution technique. Proper Riemannian metrics on the real number field can make the distribution of return rates of the stock induced by our model have the character of leptokurtosis and fat-tail; in addition, they can also explain option pricing bias and implied volatility smile (skew).

• Nonlinear Functional Analysis and Applications
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• 제29권1호
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• pp.35-45
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• 2024

In this paper, we study the existence and multiplicity of nontrivial solutions for a p-Kirchhoff equation involving critical Sobolev-Hardy exponent by using variational methods and we need to estimate the energy levels.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권3호
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• pp.253-266
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• 2019

In this article, a systematic solution based on the sequence of expansion method is planned to solve the time-fractional diffusion equation, time-fractional telegraphic equation and time-fractional wave equation in three dimensions using a current and valid approximate method, namely the ADM, VIM, and the NIM subject to the estimate initial condition. By using these three methods it is likely to find the exact solutions or a nearby approximate solution of fractional partial differential equations. The exactness, efficiency, and convergence of the method are demonstrated through the three numerical examples.

• Architectural research
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• 제13권1호
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• pp.17-24
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• 2011

Isogeometric solution of Poisson equation is provided. NURBS (NonUniform B-spline Surface) is introduced to express both geometry of structure and unknown field of governing equation. The terms of stiffness matrix and load vector are consistently derived with very accurate geometric definition. The validity of the isogeometric formulation is demonstrated by using two numerical examples such as square plate and L-shape plate. From numerical results, the present solutions have a good agreement with analytical and finite element (FE) solutions with the use of a few cells in isogeometric analysis.

• East Asian mathematical journal
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• 제38권5호
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• pp.603-610
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• 2022

In [6, 7] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. They considered a partial differential equation with the input function f ∈ L²(Ω). In this paper we consider a PDE with the input function f ∈ H¹(Ω) and find the corresponding singular and dual singular functions. We also induce the corresponding extraction formula which are the basic element for the approach.

• Nuclear Engineering and Technology
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• 제55권6호
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• pp.2305-2314
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• 2023

The governing equations of atmospheric dispersion most often taking the form of a second-order partial differential equation (PDE). Currently, typical computational codes for predicting atmospheric dispersion use the Gaussian plume model that is an analytic solution. A Gaussian model is simple and enables rapid simulations, but it can be difficult to apply to situations with complex model parameters. Recently, a method of solving PDEs using artificial neural networks called physics-informed neural network (PINN) has been proposed. The PINN assumes the latent (hidden) solution of a PDE as an arbitrary neural network model and approximates the solution by optimizing the model. Unlike a Gaussian model, the PINN is intuitive in that it does not require special assumptions and uses the original equation without modifications. In this paper, we describe an approach to atmospheric dispersion modeling using the PINN and show its applicability through simple case studies. The results are compared with analytic and fundamental numerical methods to assess the accuracy and other features. The proposed PINN approximates the solution with reasonable accuracy. Considering that its procedure is divided into training and prediction steps, the PINN also offers the advantage of rapid simulations once the training is over.

• International Journal of Aeronautical and Space Sciences
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• 제18권4호
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• pp.816-826
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• 2017

In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

• 대한수학회지
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• 제41권6호
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• pp.1049-1070
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• 2004

We classify all partial differential equations with polynomial coefficients in $\chi$ and y of the form A($\chi$) $u_{{\chi}{\chi}}$ ＋ 2B($\chi$, y) $u_{{\chi}y}$ ＋ C(y) $u_{yy}$ ＋ D($\chi$) $u_{{\chi}}$ ＋ E(y) $u_{y}$ = λu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.s.

• 물과 미래
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• 제22권2호
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• pp.233-239
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• 1989

부정류지하수 흐름에 대한 편미분방정식 Blotzmann 변환을 통하여 상미분방정식으로 변환되었으며 유한차분법을 이용하여 수치해를 구하였다. Richardson법과 차분식을 이용하여 미지초기수면구배(missing intial slope)를 구하는 새로운 방법이 제안되었다. 본 연구에서 제안된 방법으로 초기수면구배를 구하였으며 이 값들은 다른 연구결과와 비교한 바 아주 좋은 일치를 보여주었으며 또한 이 방법이 해를 구하는데 간편하고 쉬운 방법임을 보여주었다.