• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.83-98
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• 2024

In this paper, we investigate the existence and uniqueness of solutions to a new class of integro-differential equation boundary value problems (BVPs) with ㄒ-Hilfer operator. Our problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed points coincide with the solutions to the given problem. Using Banach's and Schauder's fixed point techniques, the uniqueness and existence result for the given problem are demonstrated. The stability results for solutions of the given problem are also discussed. In the end. One example is provided to demonstrate the obtained results

• Journal of the Korean Data and Information Science Society
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• v.25 no.4
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• pp.915-920
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• 2014

We consider a continuous time surplus process with investments the sizes of which are independent and identically distributed. It is assumed that an investment of the surplus to other business is made, if and only if the surplus reaches a given sufficient level. We establish an integro-differential equation for the distribution function of the surplus and solve the equation to obtain the moment generating function for the stationary distribution of the surplus. As a consequence, we obtain the first and second moments of the level of the surplus in an infinite horizon.

• 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.

• Communications for Statistical Applications and Methods
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• v.23 no.5
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• pp.423-432
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• 2016

In this paper, we stochastically analyze the continuous time surplus process in a risk model which involves a continuous type investment. It is assumed that the investment of the surplus to other business is continuously made at a constant rate, while the surplus process stays over a given sufficient level. We obtain the stationary distribution of the surplus level and/or its moment generating function by forming martingales from the surplus process and applying the optional sampling theorem to the martingales and/or by establishing and solving an integro-differential equation for the distribution function of the surplus level.

• Communications for Statistical Applications and Methods
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• v.20 no.6
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• pp.475-480
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• 2013

The surplus process in a risk model is stochastically analyzed. We obtain the characteristic function of the level of the surplus at a finite time, by establishing and solving an integro-differential equation for the distribution function of the surplus. The characteristic function of the stationary distribution of the surplus is also obtained by assuming that an investment of the surplus is made to other business when the surplus reaches a sufficient level. As a consequence, we obtain the first and second moments of the surplus both at a finite time and in an infinite horizon (in the long-run).

• Journal for History of Mathematics
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• v.16 no.2
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• pp.117-128
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• 2003

We investigate the origin and recent history for fuzzy equations. And we introduce the existence theorems of solutions for the fuzzy differential equation with infinite delays and fuzzy functional integral equations. We will also recent researches for controllability of sobolev-type semilinear integro-differential fuzzy system.

• Structural Engineering and Mechanics
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• v.4 no.3
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• pp.227-242
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• 1996

An important class of problems in the field of geotechnical engineering may be analyzed with the aid of a simple integro-differential equation. Behavior of "rigid" piles(say concrete piles), "deformable" piles(say gravel piles), pile groups, pile-raft foundations, heavily reinforced earth, flow within circular silos and down drag on cylindrical structures (for example the crusher unit of a mineral processing complex) are the type of situations that can be handled by this type of equation. The equation under consideration has the form; $$\frac{{\partial}w(r,\;z)}{{\partial}z}+f(z){\int}^z_0g({\xi})(\frac{{\partial}^2w(r,\;{\xi})}{{\partial}r^2}+\frac{1}{r}\frac{{\partial}w(r,\;{\xi})}{{\partial}r})d{\xi}+h(r,\;z)=0$$ where w(r, z) is the vertical displacement of a soil particle expressed as a function of the polar cylindrical space coordinates (r, z) and the symbols f, g and h represent soil properties and the loading conditions. The merit of the analysis is its simplicity (both in concept and in application) and the ease with which it can be expressed in a computer code. In the present paper the analysis is applied to investigate the behavior of a single rigid pile to bedrock. The emphasis, however, is placed on developing the equation, the numerical techique used in its evaluation and validation of the technique, hereafter called the ID technique, against a formal program, CRISP, which uses the FEM.

• Computational Structural Engineering
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• v.8 no.4
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• pp.57-64
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• 1995

This paper introduced a simple numerical analysis algorithm for the calculation of the dynamic responses of the beam structure with a moving mass. The dynamic equation of motion of the Bernoulli-Euler beam is derived by considering the moving mass as a moving particle, and the dynamic equation of motion is transformed into an integro-differential equation by use of the structural influence function. The numerical solutions of the integro-differential equation are obtained by the modal analysis approach, and compared with those cited from well-known references. The proves that the numerical analysis algorithm proposed herein provide very reliable results, and thus it can be utilized in the design analysis of the beamlike structures exited by a mass which is traveling on it.

• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.57-69
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• 2012

In this paper, we prove the existence of mild solutions for a first order impulsive neutral differential inclusion with state dependent delay. We assume that the state-dependent delay part generates an analytic resolvent operator and transforms it into an integral equation. By using a fixed point theorem for condensing multi-valued maps, a main existence theorem is established.

• 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.