• Steel and Composite Structures
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• v.37 no.5
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• pp.551-569
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• 2020

Vibration of vertically aligned-monolayered-nonuniform nanorods consist of functionally graded materials with elastic supports has not been investigated yet. To fill this gap, the problem is examined using the elasticity theories of Eringen and Gurtin-Murdoch. The geometrical and mechanical properties of the surface layer and the bulk are allowed to vary arbitrarily across the length. The nonlocal-surface energy-based governing equations are established using differential-type and integro-type formulations, and solved by employing the Galerkin method by exploiting admissible modes approach and element-free Galerkin (EFG). Through various comparison studies, the effectiveness of the EFG in capturing both nonlocal-differential/integro-based frequencies is proved. A constructive parametric study is also conducted, and the roles of nanorods' diameter, length, stiffness of both inter-rod's elastic layer and elastic supports, power-law index of both constituent materials and geometry, nonlocal and surface effects on the dominant frequencies are revealed.

• Honam Mathematical Journal
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• v.45 no.1
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• pp.54-70
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• 2023

In this article, we state and prove several new retarded nonlinear integral and integro-differential inequalities of Gronwall-Bellman-Pachpatte type. These inequalities generalize some former famous inequalities and can be used in examining the existence, uniqueness, boundedness, stability, asymptotic behaviour, quantitative and qualitative properties of solutions of nonlinear differential and integral equations. Applications are provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problem for nonlinear integro-differential equation and Volterra type retarded nonlinear equation. This research work will ensure to open the new opportunities for studying of nonlinear dynamic inequalities on time scale structure of varying nature.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.3
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• pp.189-200
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• 2010

A numerical method is proposed and analyzed to approximate a mathematical model of age-dependent population dynamics with spatial diffusion. The model takes a form of nonlinear and nonlocal system of integro-differential equations. A finite difference method along the characteristic age-time direction is considered and primal mixed finite elements are used in the spatial variable. A priori error estimates are derived for the relevant variables.

• The Korean Journal of Applied Statistics
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• v.25 no.5
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• pp.813-820
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• 2012

A surplus process with two types of claims is considered, where Type I claims occur more frequently, however, their sizes are smaller stochastically than Type II claims. The ruin probabilities of the surplus caused by each type of claim are obtained by establishing integro-differential equations for the ruin probabilities. The formulas of the ruin probabilities contain an infinite sum and convolutions that make the formulas hard to be applicable in practice; subsequently, we obtain explicit formulas for the ruin probabilities when the sizes of both types of claims are exponentially distributed. Finally, we show through a numerical example, that Type II claims have more impact on the ruin probability of the surplus than Type I claims.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.939-955
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• 2022

In this paper, we study a first-order non-linear singularly perturbed Volterra integro-differential equation (SPVIDE). We discretize the problem by a uniform difference scheme on a Bakhvalov-Shishkin mesh. The scheme is constructed by the method of integral identities with exponential basis functions and integral terms are handled with interpolating quadrature rules with remainder terms. An effective quasi-linearization technique is employed for the algorithm. We establish the error estimates and demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N^-1) uniformly convergent, where N is the mesh parameter. The numerical results on a couple of examples are also provided to confirm the theoretical analysis.

• Journal of computational fluids engineering
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• v.3 no.1
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• pp.11-21
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• 1998

A vorticity-based method for the numerical solution of the two-dimensional incompressible Navier-Stokes equations is presented. The governing equations for vorticity, velocity and pressure variables are expressed in an integro-differential form. The global coupling between the vorticity and the pressure boundary conditions is fully considered in an iterative procedure when numerical schemes are employed. The finite volume method of the second order TVD scheme is implemented to integrate the vorticity transport equation with the dynamic vorticity boundary condition. The velocity field is obtained by using the Biot-Savart integral. The Green's scalar identity is used to solve the total pressure in an integral approach similar to the surface panel methods which have been well established for potential flow analysis. The present formulation is validated by comparison with data from the literature for the two-dimensional cavity flow driven by shear in a square cavity. We take two types of the cavity now: (ⅰ) driven by non-uniform shear on top lid and body forces for which the exact solution exists, and (ⅱ) driven only by uniform shear (of the classical type).

• Advances in aircraft and spacecraft science
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• v.7 no.3
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• pp.215-228
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• 2020

As is shown in the paper, the Koltunov-Rzhanitsyn singular kernel of heredity (when constructing mathematical models of the dynamics problem of the hereditary theory of viscoelasticity) adequately describes real mechanical processes, best approximates experimental data for a long period of time. A mathematical model of the problem of the flutter of viscoelastic plates moving in a gas with a high supersonic velocity is given. Using the Bubnov-Galerkin method, discrete models of the problem of the flatter of viscoelastic plates flowed over by supersonic gas flow are obtained. A numerical method is developed to solve nonlinear integro-differential equations (IDE) for the problem of the hereditary theory of viscoelasticity with weakly singular kernels. A general computational algorithm and a system of application programs have been developed, which allow one to investigate the nonlinear dynamic problems of the hereditary theory of viscoelasticity with weakly singular kernels. On the basis of the proposed numerical method and algorithm, nonlinear problems of the flutter of viscoelastic plates flowed over in a gas flow at an arbitrary angle are investigated. In a wide range of changes in various parameters of the plate, the critical velocity of the flutter is determined. It is shown that the singularity parameter α affects not only the oscillations of viscoelastic systems, but the critical velocity of the flutter as well.

• Steel and Composite Structures
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• v.46 no.6
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• pp.759-772
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• 2023

This manuscript presents a comprehensive mathematical model to investigate buckling stability and postbuckling response of bio-inspired composite beams with helicoidal orientations. The higher order shear deformation theory as well as the Timoshenko beam theories are exploited to include the shear influence. The equilibrium nonlinear integro-differential equations of helicoidal composite beams are derived in detail using the energy conservation principle. Differential integral quadrature method (DIQM) is employed to discretize the nonlinear system of differential equations and solve them via the Newton iterative method then obtain the response of helicoidal composite beam. Numerical calculations are carried out to check the validity of the present solution methodology and to quantify the effects of helicoidal rotation angle, elastic foundation constants, beam theories, geometric and material properties on buckling, postbuckling of bio-inspired helicoidal composite beams. The developed model can be employed in design and analysis of curved helicoidal composite beam used in aerospace and naval structures.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1269-1283
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• 2018

We are concerned with elliptic equations in ${\mathbb{R}}^N$, driven by a non-local integro-differential operator, which involves the fractional Laplacian. The main aim of this paper is to prove the existence of small solutions for our problem with negative energy in the sense that the sequence of solutions converges to 0 in the $L^{\infty}$-norm by employing the regularity type result on the $L^{\infty}$-boundedness of solutions and the modified functional method.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1993.04b
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• pp.175-180
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• 1993

The nonlinear integro-differential equations of motion of a two-link structurally flexible planar manipulator executing contact tasks are presented. The equations of motion are derived using the extended Hamilton's principle and the Galerkin criterion. Also, Models for the wrist-force sensor and impact that occurs when the manipulator's end point makes contact withthe environment are presented. The dynamic models presented can be used to studythe dynamics of the system and to design controllers.