• Title/Summary/Keyword: partial differential-difference equations

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ON MEROMORPHIC SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS OF FIRST ORDER IN SEVERAL COMPLEX VARIABLES

  • Qibin Cheng;Yezhou Li;Zhixue Liu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.2
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    • pp.425-441
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    • 2023
  • This paper is concerned with the value distribution for meromorphic solutions f of a class of nonlinear partial differential-difference equation of first order with small coefficients. We show that such solutions f are uniquely determined by the poles of f and the zeros of f - c, f - d (counting multiplicities) for two distinct small functions c, d.

SOLUTIONS FOR QUADRATIC TRINOMIAL PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS IN ℂn

  • Molla Basir Ahamed;Sanju Mandal
    • Journal of the Korean Mathematical Society
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    • v.61 no.5
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    • pp.975-995
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    • 2024
  • In this paper, we utilize Nevanlinna theory to study the existence and forms of solutions for quadratic trinomial complex partial differential-difference equations of the form aF2 + 2ωFG + bG2 = exp(g), where ab ≠ 0, ω ∈ ℂ with ω2 ≠ 0, ab and g is a polynomial in ℂn. In order to achieve a comprehensive and thorough analysis, we study the characteristics of solutions in two specific cases: one when ω2 ≠ 0, ab and the other when ω = 0. Because polynomials in several complex variables may exhibit periodic behavior, a property that differs from polynomials in single complex variables, our study of finding solutions of equations in ℂn is significant. The main results of the paper improved several known results in ℂn for n ≥ 2. Additionally, the corollaries generalize results of Xu et al. [Rocky Mountain J. Math. 52(6) (2022), 2169-2187] for trinomial equations with arbitrary coefficients in ℂn. Finally, we provide examples that endorse the validity of the conclusions drawn from the main results and their related remarks.

A Generalized Finite Difference Method for Solving Fokker-Planck-Kolmogorov Equations

  • Zhao, Li;Yun, Gun Jin
    • International Journal of Aeronautical and Space Sciences
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    • v.18 no.4
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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.

SOLVING PARTIAL DIFFERENTIAL ALGEBRAIC EQUATIONS BY COLLOCATION AND RADIAL BASIS FUNCTIONS

  • Bao, Wendi;Song, Yongzhong
    • Journal of applied mathematics & informatics
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    • v.30 no.5_6
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    • pp.951-969
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    • 2012
  • In this paper, we propose a class of meshless collocation approaches for the solution of time dependent partial differential algebraic equations (PDAEs) in terms of a radial basis function interpolation numerical scheme. Kansa's method and the Hermite collocation method (HCM) for PDAEs are given. A sensitivity analysis of the solutions from different shape parameter c is obtained by numerical experiments. With use of the random collocation points, we have obtain the more accurate solution by the methods than those by the finite difference method for the PDAEs with index-2, i.e, we avoid the influence from an index jump of PDAEs in some degree. Several numerical experiments show that the methods are efficient.

UNIFORMLY CONVERGENT NUMERICAL SCHEME FOR A SINGULARLY PERTURBED DIFFERENTIAL-DIFFERENCE EQUATIONS ARISING IN COMPUTATIONAL NEUROSCIENCE

  • DABA, IMIRU TAKELE;DURESSA, GEMECHIS FILE
    • Journal of applied mathematics & informatics
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    • v.39 no.5_6
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    • pp.655-676
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    • 2021
  • A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.

Numerical Solution of Second Order Linear Partial Differential Equations using Agricultural Systems Application Platform (농업시스템응용플랫폼을 이용한 2계 편미분 방정식의 해석)

  • Lee, SungYong;Kim, Taegon;Suh, Kyo;Han, Yicheol;Lee, Jemyung;Yi, Hojae;Lee, JeongJae
    • Journal of The Korean Society of Agricultural Engineers
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    • v.58 no.1
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    • pp.81-90
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    • 2016
  • The Agricultural Systems Application Platform (ASAP) provides bottom-up modelling and simulation environment for agricultural engineer. The purpose of this study is to expand usability of the ASAP to the second order partial differential equations: elliptic equations, parabolic equations, and hyperbolic equations. The ASAP is a general-purpose simulation tool which express natural phenomenon with capsulized independent components to simplify implementation and maintenance. To use the ASAP in continuous problems, it is necessary to solve partial differential equations. This study shows usage of the ASAP in elliptic problem, parabolic problem, and hyperbolic problem, and solves of static heat problem, heat transfer problem, and wave problem as examples. The example problems are solved with the ASAP and Finite Difference method (FDM) for verification. The ASAP shows identical results to FDM. These applications are useful to simulate the engineering problem including equilibrium, diffusion and wave problem.

AN ACCURATE AND EFFICIENT NUMERICAL METHOD FOR BLACK-SCHOLES EQUATIONS

  • Jeong, Da-Rae;Kim, Jun-Seok;Wee, In-Suk
    • Communications of the Korean Mathematical Society
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    • v.24 no.4
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    • pp.617-628
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    • 2009
  • We present an efficient and accurate finite-difference method for computing Black-Scholes partial differential equations with multiunderlying assets. We directly solve Black-Scholes equations without transformations of variables. We provide computational results showing the performance of the method for two underlying asset option pricing problems.

A new approach for finite element analysis of delaminated composite beam, allowing for fast and simple change of geometric characteristics of the delaminated area

  • Perel, Victor Y.
    • Structural Engineering and Mechanics
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    • v.25 no.5
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    • pp.501-518
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    • 2007
  • In this work, a new approach is developed for dynamic analysis of a composite beam with an interply crack, based on finite element solution of partial differential equations with the use of the COMSOL Multiphysics package, allowing for fast and simple change of geometric characteristics of the delaminated area. The use of COMSOL Multiphysics package facilitates automatic mesh generation, which is needed if the problem has to be solved many times with different crack lengths. In the model, a physically impossible interpenetration of the crack faces is prevented by imposing a special constraint, leading to taking account of a force of contact interaction of the crack faces and to nonlinearity of the formulated boundary value problem. The model is based on the first order shear deformation theory, i.e., the longitudinal displacement is assumed to vary linearly through the beam's thickness. The shear deformation and rotary inertia terms are included into the formulation, to achieve better accuracy. Nonlinear partial differential equations of motion with boundary conditions are developed and written in the format acceptable by the COMSOL Multiphysics package. An example problem of a clamped-free beam with a piezoelectric actuator is considered, and its finite element solution is obtained. A noticeable difference of forced vibrations of the delaminated and undelaminated beams due to the contact interaction of the crack's faces is predicted by the developed model.

The Effect of Neglecting the Longitudinal Moment Terms on the Natural Frequency of Laminated Plates with Increasing Aspect Ratio (보강재 보강 형태에 따른 특별직교 이방성 적층복합판의 고유진동수에 대한 종방향 모멘트 무시효과)

  • 김덕현;김경진;이정호;박정호
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • 1998.10a
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    • pp.109-116
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    • 1998
  • The method of vibration analysis used is the one developed by the senior author. He developed and reported, in 1974, a simple but exact method of calculating the natural frequency of beam and tower structures with irregular cross-sections and attached mass/masses. Since 1989, this method has been extended to two-dimensional problems with several types of given conditions and has been reported at several international conferences. This method uses the deflection influence surfaces. The finite difference method is used for this purpose, in this paper. In order to reduce the pivotal points required, the three simultaneous partial differential equations of equilibrium with three dependent variables, w, M$_{x}$, and $M_{y}$, are used instead of the one forth order partial differential equation. By neglecting the M$_{x}$ terms, the size of the matrices needed to solve the resulting linear equations are reduced to two thirds of the "non-modified" equations.tions.

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A Study on Natural Convection from Two Cylinders in a Cavity

  • Mochimaru Yoshihiro;Bae Myung-Whan
    • Journal of Mechanical Science and Technology
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    • v.20 no.10
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    • pp.1773-1778
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    • 2006
  • Steady-state natural convection heat transfer characteristics from cylinders in a multiply-connected bounded region are clarified. A spectral finite difference scheme (spectral decomposition of the system of partial differential equations, semi-implicit time integration) is applied in numerical analysis, with a boundary-fitted conformal coordinate system through a Jacobian elliptic function with a successive transformation to formulate a system of governing equations in terms of a stream function, vorticity and temperature. Multiplicity of the domain is expressed explicitly.