• Journal of Mechanical Science and Technology
• /
• v.18 no.8
• /
• pp.1375-1387
• /
• 2004

A solution is given for the elastodynamic problem of a crack perpendicular to the graded interfacial zone in bonded materials under the action of anti plane shear impact. The interfacial zone is modeled as a nonhomogeneous interlayer with the power-law variations of its shear modulus and mass density between the two dissimilar, homogeneous half-planes. Laplace and Fourier integral transforms are employed to reduce the transient problem to the solution of a Cauchy-type singular integral equation in the Laplace transform domain. Via the numerical inversion of the Laplace transforms, the values of the dynamic stress intensity factors are obtained as a function of time. As a result, the influences of material and geometric parameters of the bonded media on the overshoot characteristics of the dynamic stress intensities are discussed. A comparison is also made with the corresponding elastostatic solutions, addressing the inertia effect on the dynamic load transfer to the crack tips for various combinations of the physical properties.

• Journal of the Korean Mathematical Society
• /
• v.49 no.5
• /
• pp.1065-1082
• /
• 2012

In this paper we first investigate the existence of the generalized Fourier-Feynman transform of the functional F given by $$F(x)={\hat{\nu}}((e_1,x)^{\sim},{\ldots},(e_n,x)^{\sim})$$, where $(e,x)^{\sim}$ denotes the Paley-Wiener-Zygmund stochastic integral with $x$ in a very general function space $C_{a,b}[0,T]$ and $\hat{\nu}$ is the Fourier transform of complex measure ${\nu}$ on $B({\mathbb{R}}^n)$ with finite total variation. We then define two sequential transforms. Finally, we establish that the one is to identify the generalized Fourier-Feynman transform and the another transform acts like an inverse generalized Fourier-Feynman transform.

• Structural Engineering and Mechanics
• /
• v.30 no.5
• /
• pp.577-592
• /
• 2008

Laplace-Fourier transform techniques are used to investigate the interaction caused by mechanical, thermal and microstress sources in a generalized thermomicrostretch elastic medium with temperature-dependent mechanical properties. The modulus of elasticity is taken as a linear function of reference temperature. The integral transforms are inverted using a numerical technique to obtain the normal stress, tangential stress, tangential couple stress, microstress and temperature distribution. Effect of temperature dependent modulus of elasticity and thermal relaxation times have been depicted graphically on the resulting quantities. Comparisons are made with the results predicted by the theories of generalized thermoelasticity. Some particular cases are also deduced from the present investigation.

• Communications of the Korean Mathematical Society
• /
• v.33 no.4
• /
• pp.1159-1170
• /
• 2018

Recently, Parmar [5] introduced a new extension of the ${\tau}$-Gauss hypergeometric function $_2R^{\tau}_1(z)$. The main object of this paper is to study this extended ${\tau}$-Gauss hypergeometric function and obtain its properties including connection with modified Bessel function of third kind and extended generalized hypergeometric function, several contiguous relations, differential relations, integral transforms and elementary integrals. Various special cases of our results are also discussed.

• Communications of the Korean Mathematical Society
• /
• v.37 no.4
• /
• pp.1055-1072
• /
• 2022

Our aim is to establish certain image formulas of the (p, 𝜈)-extended Gauss' hypergeometric function F_p,𝜈(a, b; c; z) by using Saigo's hypergeometric fractional calculus (integral and differential) operators. Corresponding assertions for the classical Riemann-Liouville(R-L) and Erdélyi-Kober(E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, 𝜈)-extended Gauss's hypergeometric function F_p,𝜈(a, b; c; z) and Fox-Wright function _rΨ_s(z). We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the (p, 𝜈)-extended Gauss' hypergeometric function F_p,𝜈(a, b; c; z).

• Structural Engineering and Mechanics
• /
• v.20 no.2
• /
• pp.209-223
• /
• 2005

A new method for deriving analytical solution of the annular elastic plate on elastic foundation under axisymmetric loading is presented. The formulation is based on application of Hankel integral transforms and Bessel functions' properties in the corresponding boundary-value problem. A representative example is studied and the obtained solution is compared with published numerical results indicating excellent agreement.

• Korean Journal of Mathematics
• /
• v.6 no.1
• /
• pp.47-66
• /
• 1998

In this paper, we introduce an $L_p$ analytic Fourier-Feynman transformation, show the existence of the $L_p$ analytic Fourier-Feynman transforms for a certain class of cylinder functionals on an abstract Wiener space, and investigate its interesting properties. Moreover, we define a convolution product for two functionals on the abstract Wiener space and establish the relationships between the Fourier-Feynman transform for the convolution product of two cylinder functionals and the Fourier-Feynman transform for each functional.

• Communications of the Korean Mathematical Society
• /
• v.11 no.3
• /
• pp.707-723
• /
• 1996

We first define the concept of the generalized analytic Fourier-Feynman transforms of a class of functionals on function space induced by a generalized Brownian motion process and study of functionals which plays on important role in physical problem of the form $ F(x) = {\int^{T}_{0} f(t, x(t))dt} $ where f is a complex-valued function on $[0, T] \times R$. We next show that the generalized analytic Fourier-Feynman transform of the convolution product is a product of generalized analytic Fourier-Feynman transform of functionals on functin space.

• Journal of the Korean Mathematical Society
• /
• v.49 no.4
• /
• pp.829-842
• /
• 2012

In this paper, we give the condition for the boundedness of the Berezin transforms on Herz spaces with a normal weight on the unit ball of $\mathbb{C}^n$. And we provide the integral estimates concerning pluriharmonic kernel functions. Using this, we finally obtain the growth estimates of the Berezin transforms on such Herz spaces.

• Communications of the Korean Mathematical Society
• /
• v.35 no.3
• /
• pp.809-823
• /
• 2020

Let C[0, T] denote the space of continuous, real-valued functions on the interval [0, T] and let C₀[0, T] be the space of functions x in C[0, T] with x(0) = 0. In this paper, we introduce a Banach algebra ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ on C[0, T] and its equivalent space ${\bar{\mathcal{F}}}({\mathcal{H}}) $, a space of transforms of equivalence classes of measures, which generalizes Fresnel class 𝓕(𝓗), where 𝓗 is an appropriate real separable Hilbert space of functions on [0, T]. We also investigate their properties and derive an isomorphism between ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ and ${\bar{\mathcal{F}}}({\mathcal{H}}) $. When C[0, T] is replaced by C₀[0, T], ${\bar{\mathcal{F}}}({\mathcal{H}}) $ and ${\bar{\mathcal{S}}}_{{\alpha},{\beta};{\varphi}}$ reduce to 𝓕(𝓗) and Cameron-Storvick's Banach algebra 𝓢, respectively, which is the space of generalized Fourier-Stieltjes transforms of the complex-valued, finite Borel measures on L²[0, T].