Numerical Approximation Methods

Cohen, Harold

doi:10.1007/978-1-4419-9837-8

Cited by 38 publications

(26 citation statements)

References 17 publications

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“…Because resulted curves of Fig. 4 looked like to the curve of log k n, so we tried a least square curve fitting method [40], to fit the outcomes of equation No. 7's analytical results in the equation below:…”

Section: Resultsmentioning

confidence: 99%

A Novel Probabilistic Key Management Algorithm for Large-Scale MANETs

Gharib

Emamjomeh-Zadeh

Norouzi-Fard

et al. 2013

2013 27th International Conference on Advanced Information Networking and Applications Workshops

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Mobile ad hoc networks (MANETs) have become very interesting during last years, but the security is the most important problem they suffer from. Asymmetric cryptography is a very useful solution to provide a secure environment in multihop networks where intermediate nodes are able to read, drop or change messages before resending them. However, storing all keys in every node by this approach is inefficient, if practically possible, in large-scale MANETs due to some limitations such as memory or process capability. In this paper, we propose a new probabilistic key management algorithm for large-scale MANETs. To the best of our knowledge, this is the first method which probabilistically uses asymmetric cryptography to manage the keys in MANETs. In this algorithm, we store only a few keys in each node instead of all. We analytically prove that the network will remain connected with a high probability more than 99.99%. Furthermore, we analytically calculate the average path length in the network and show that this parameter will not have a significant increment using our algorithm. All analytical results are also validated by simulation to make them dependable.

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Section: Resultsmentioning

confidence: 99%

A Novel Probabilistic Key Management Algorithm for Large-Scale MANETs

Gharib

Emamjomeh-Zadeh

Norouzi-Fard

et al. 2013

2013 27th International Conference on Advanced Information Networking and Applications Workshops

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show abstract

“…When P i < P ij , the integral in Equation () is undefined. Therefore, we solve this integral by numerical solutions such as Gauss‐Legendre quadrature method . Firstly, we use

k = \tan ((), \frac{π}{4} ((), 1 + ω))

to change

I = \int_{0}^{\infty} g ((), k) italicdk

\frac{π}{4} \int_{- 1}^{1} ((), \frac{1}{{normalcos}^{2} (\frac{π}{4} (1 + ω))}) g ((), \tan ((), \frac{π}{4} ((), 1 + ω))) d ω

.…”

Section: Outage Probability For No‐ib Relay‐aided D2d Communicationsmentioning

confidence: 99%

“…In Equation (), ω k is the root of Legendre quadrature ( P n ( ω k ) = 0) which is given in Equation () and n is the number of points. The weights are as Equation () . Equation () shows the Gauss‐Legendre‐based solution for integral I .…”

Section: Outage Probability For No‐ib Relay‐aided D2d Communicationsmentioning

confidence: 99%

Outage analysis of nonorthogonal in‐band relay‐aided device‐to‐device communications in Rayleigh fading channels

Moghaddam

2017

Internet Technology Letters

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Minimizing the outage probability and/or maximizing the throughput, optimally realize relay‐aided nonorthogonal in‐band (NO‐IB) device‐to‐device (D2D) communications. In this letter, we derive closed‐form expressions for the outage probability of NO‐IB amplify‐and‐forward (AF) and decode‐and‐forward (DF) relay‐aided D2D communications in Rayleigh fading channels. Numerical analyses show that the closed‐form formulas are in close agreement with the simulation results. Furthermore, we investigate the effect of signal‐to‐noise ratio and the distances between D2D pairs and relay nodes as well as cellular users. First results show that in the low‐noise regime, AF outperforms DF while they perform very close in the moderate noise level and DF outperforms AF in the high‐noise power. Second, we indicate that the relay‐aided AF/DF D2D communications is more reliable than the direct one when the TX/RX nodes of a D2D pair is distant approximately more than half of the cell radius.

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“…Particularly, in previous studies, they are used to describe the problems of long‐term forecasting and control of groundwater level in the soil moisture. Various problems for these equations and methods of solving them are studied in the literature …”

Section: Introductionmentioning

confidence: 99%

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

Dzhumabaev

2017

Math Methods in App Sciences

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The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.

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Numerical Approximation Methods

Cited by 38 publications

References 17 publications

A Novel Probabilistic Key Management Algorithm for Large-Scale MANETs

A Novel Probabilistic Key Management Algorithm for Large-Scale MANETs

Outage analysis of nonorthogonal in‐band relay‐aided device‐to‐device communications in Rayleigh fading channels

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

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