• 호남수학학술지
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• 제24권1호
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• pp.67-73
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• 2002

In this paper we discuss on the existence of general solution of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z})+G(z,\;w,\;\bar{w})$ in the Sololev Space $W_{1,p}(D)$, that is generalization of a first order Non-linear Elliptic System of Partial Differential Equations $\frac{{\partial}w}{{\partial}\bar{z}}=F(z,\;w,\;\frac{{\partial}w}{{\partial}z}).$

• 호남수학학술지
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• 제27권2호
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• pp.205-209
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• 2005

In this paper we discuss on the uniquenss of the solution of partial differential equations: (1) $${\frac{{\partial}w}{{\partial}{\bar{z}}}}=F(z,\;w,\;{\frac{{\partial}w}{{\partial}z}}}+G(z,\;w.{\bar{w})$$ in the sobolev space $W_{1,p}(D)$.

• Journal of applied mathematics & informatics
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• 제4권1호
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• pp.109-116
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• 1997

In the present paper we consider an abstract partial dif-ferential equation of the form $\frac{\partial^2u}{{\partial}t^2}-\frac{\partial^2u}{{\partial}x^2}+A(x.t)u=f(x, t)$, where ${A(x, t):(x, t){\epsilon}\bar{G} }$ is a family of linear closed operators and $G=GU{\partial}G$, G is a suitable bounded region in the (x, t)-plane with bound-are ${\partial}G$. It is assumed that u is given on the boundary ${\partial}G$. The objective of this paper is to study the considered Dirichlet problem for a wide class of operators $A(x, t)$. A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considerde.

• Computers and Concrete
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• 제7권4호
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• pp.317-329
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• 2010

The paper describes localization of deformation in a bar under tensile loading. The material of the bar is considered as non-linear viscous elastic and the bar consists of two symmetric halves. It is assumed that the model represents behavior of the quasi-brittle viscous material under uniaxial tension with different loading rates. Besides that, the bar could represent uniaxial stress-strain law on a single plane of a microplane material model. Non-linear material property is taken from the microplane material model and it is coupled with the viscous damper producing non-linear Maxwell material model. Mathematically, the problem is described with a system of two partial differential equations with a non-linear algebraic constraint. In order to obtain solution, the system of differential algebraic equations is transformed into a system of three partial differential equations. System is subjected to loadings of different rate and it is shown that localization occurs only for high loading rates. Mathematically, in such a case two solutions are possible: one without the localization (unstable) and one with the localization (stable one). Furthermore, mass is added to the bar and in that case the problem is described with a system of four differential equations. It is demonstrated that for high enough loading rates, it is the added mass that dominates the response, in contrast to the viscous and elastic material parameters that dominated in the case without mass. This is demonstrated by several numerical examples.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.31-48
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• 2007

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable function $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.

• Structural Engineering and Mechanics
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• 제85권4호
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• pp.545-562
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• 2023

In this work, superharmonic and subharmonic resonance of rotating stiffened FGM truncated conical shells exposed to harmonic excitation in a thermal environment is investigated. Utilizing classical shell theory considering Coriolis acceleration and the centrifugal force, the governing equations are extracted. Non-linear model is formulated employing the von Kármán non-linear relations. In this study, to model the stiffener effects the smeared stiffened technique is utilized. The non-linear partial differential equations are discretized into non-linear ordinary differential equations by applying Galerkin's method. The method of multiple scales is utilized to examine the non-linear superharmonic and subharmonic resonances behavior of the conical shells. In this regard, the effects of the rotating speed of the shell on the frequency response plot are investigated. Also, the effects of different semi-vertex angles, force amplitude, volume-fraction index, and temperature variations on the frequency-response graph are examined for different rotating speeds of the stiffened FGM truncated conical shells.

• Advances in nano research
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• 제12권1호
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• pp.37-47
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• 2022

In this paper, the non-linear static analysis of Timoshenko nanobeams consisting of bi-directional functionally graded material (BFGM) with immovable ends is investigated. The scratching in the FG nanobeam mid-plane, is the source of nonlinearity of the bending problems. The nonlocal theory is used to investigate the non-linear static deflection of nanobeam. In order to simplify the formulation, the problem formulas is derived according to the physical middle surface. The Hamilton principle is employed to determine governing partial differential equations as well as boundary conditions. Moreover, the differential quadrature method (DQM) and direct iterative method are applied to solve governing equations. Present results for non-linear static deflection were compared with previously published results in order to validate the present formulation. The impacts of the nonlocal factors, beam length and material property gradient on the non-linear static deflection of BFG nanobeams are investigated. It is observed that these parameters are vital in the value of the non-linear static deflection of the BFG nanobeam.

• Steel and Composite Structures
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• 제31권2호
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• pp.187-199
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• 2019

This paper presents the semi-analytical development of the dynamic instability behavior and the dynamic response of functionally graded (FG) cylindrical shallow shell panel subjected to different type of periodic axial compression. First, in prebuckling analysis, the stresses distribution within the panels are determined for respective loading type and these stresses are used to study the dynamic instability behavior and the dynamic response. The prebuckling stresses within the shell panel are the same as applied in-plane edge loading for the case of uniform and linearly varying loadings. However, this is not true for the case of parabolic loadings. The parabolic edge loading produces all the stresses (${\sigma}_{xx}$, ${\sigma}_{yy}$ and ${\tau}_{xy}$) within the FG cylindrical panel. These stresses are evaluated by minimizing the membrane energy via Ritz method. Using these stresses the partial differential equations of FG cylindrical panel are formulated by applying Hamilton's principal assuming higher order shear deformation theory (HSDT) and von-$K{\acute{a}}rm{\acute{a}}n$ non-linearity. The non-linear governing partial differential equations are converted into a set of Mathieu-Hill equations via Galerkin's method. Bolotin method is adopted to trace the boundaries of instability regions. The linear and non-linear dynamic responses in stable and unstable region are plotted to know the characteristics of instability regions of FG cylindrical panel. Moreover, the non-linear frequency-amplitude responses are obtained using Incremental Harmonic Balance (IHB) method.

• Computers and Concrete
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• 제34권3호
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• pp.347-353
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• 2024

This paper investigates the second-grade fluid between two parallel plates. Fluid is produced due to stretching. Convective heat and mass transfer features are elaborated with thermal and solutal stratification. Thermal radiation and chemical reactions are also assumed in heat and mass transport processes partial differential. Formulated non-linear partial differential equations are transformed into non-linear ordinary differential equations by utilizing the suitable transformation. Convergent series solutions are computed via Homotopy Analysis Method (HAM). Effects of Hartman number, temperature field, velocity distribution and Prandtl number are sketched and analyzed through graphs. It is noticed that velocity field first decreases and after some distance it shows increasing behavior by the increment.

• 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.