• Kyungpook Mathematical Journal
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• v.20 no.1
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• pp.77-82
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• 1980

There are many definitions of the fractional derivative. It is purpose of this paper to show some results which were got for fractional partial derivative of functions of two variables and to give an application of the fractional partial derivative.

• Korean Journal of Mathematics
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• v.28 no.4
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• pp.865-875
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• 2020

We provide detailed proofs for local gradient estimates for weak solutions to elliptic equations in divergence form with partial Dini mean oscillation coefficients subject to conormal derivative boundary conditions.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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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 $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.

• Korean Journal of Mathematics
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• v.31 no.3
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• pp.259-267
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• 2023

The paper presents the introduction of a novel linear derivative operator for meromorphic functions that are linked with q-calculus. Using the linear derivative operator, a new category of meromorphic functions is generated in the paper. We obtain sufficient conditions and show some properties of functions belonging to these subclasses. The partial sums of its sequence and the q-neighborhoods problem are solved.

• Geophysics and Geophysical Exploration
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• v.1 no.1
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• pp.8-18
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• 1998

In exploration seismology, the Kirchhoff hyperbola has been successfully used to migrate reflection seismo-grams. The mathematical basis of Kirchhoff hyperbola has not been clearly defined and understood for the application of prestack or poststack migration. The travel time from the scatterer in the subsurface to the receivers (exploding reflector model) on the surface can be a kinematic approximation of Green's function when the source is excited at position of the scatterer. If we add the travel time from the source to the scatterer in the subsurface to the travel time of exploding reflector model, we can view this travel time as a kinematic approximation of the partial derivative wavefield with respect to the velocity or the density in the subsurface. The summation of reflection seismogram along the Kirchhoff hyperbola can be evaluated as an inner product between the partial derivative wavefield and the field reflection seismogram. In addition to this kinematic interpretation of Kirchhoff hyperbola, when we extend this concept to shallow refraction seismic data, the stacking of refraction data along the straight line can be interpreted as a measurement of an inner product between the first arrival waveform of the partial derivative wavefield and the field refraction data. We evaluated the Kirchhoff hyperbola and the straight line for stacking the refraction data in terms of the first arrival waveform of the partial derivative wavefield with respect to the velocity or the density in the subsurface. This evaluation provides a firm and solid basis for the conventional Kirchhoff migration and the straight line stacking of the refraction data.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Clinical and Experimental Pediatrics
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• v.51 no.11
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• pp.1241-1244
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• 2008

We report on 2 siblings with a partial trisomy of 7q ($7q22{\rightarrow}qter$) and concomitant partial monosomy of 8p ($8p23.3{\rightarrow}pter$), which were shown by FISH using probes located at the telomere region of each chromosome. All the balanced translocation carriers (father and a sister) in this family had a normal phenotype. The 2 siblings with the same abnormal karyotype had similar multiple congenital anomalies and dysmorphic features. During the follow-up, the first male patient died in the neonatal period, but the female sibling is still alive at 2 years and 6 months of age.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.241-252
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• 2011

Let $\{\Omega_{\tau}\}_{\tau{\in}I}$ be a family of strictly convex domains in $\mathbb{C}^n$. We obtain explicit estimates for the solution of the $\bar{\partial}$-equation on $\Omega{\times}I$ in H$\ddot{o}$lder space. We also obtain explicit point-wise derivative estimates for the $\bar{\partial}$-equation both in space and parameter variables.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.401-416
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• 1994

We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {\Delta u = f(x,u,\Delta u), in \Omega {Bu(x) = \phi(x), on \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $R^n, n \geq 1, and \partial\Omega \in C^{2,\alpha}, (0 < \alpha < 1), \Delta$ is the Laplacian operator, $\nabla u = (D_1u, D_2u, \cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)\frac{d\nu}{du} (x), $$ where $\frac{d\nu}{du} denotes the outward normal derivative of u on $\partial\Omega$.

• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.167-178
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• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.