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The purpose of this paper is to study repairable consecutive‐k‐out‐of‐n: systems with r repairmen and a sensing device.

The system can either be a circular C(k, n: G) system or a linear C(k, n: G) system. The working time and the repair time of each component in the system and the sensor detection time are exponentially distributed. Every component after repair is perfect. Each component is classified as either a key component, or an ordinary one according to its priority role to system's repair. A sensing device is introduced to detect the failure of each component in the system in advance and completion of repair of components. If the repair is completed, the sensor will send the component to standby according to its priority. The state transition probabilities of the system are derived using the definition of generalized transition probability. To obtain the reliability and availability Laplace transform techniques have been used.

The Kolmogorov‐Feller forward equations are derived for both linear and circular systems. Reliability and MTTF of both the systems are derived using Laplace transforms. Numerical examples are given in detail to demonstrate the theoretical results and these verify the validity of the studied system.

A consecutive‐k‐out‐of‐n system consists of a sequence of n‐ordered components along a line or a circle such that the system is good if and only if at least k consecutive components in the system are good. Each component in the system is classified as key component or ordinary component according to its priority in system functioning. By using a sensing device the failure can be detected in advance.

This study indicates that by using a sensing device we can detect the failure in advance. Thus, the reliability and MTTF of the system can be improved.

This paper discusses theoretical extensions of the implied volatility method of Dupire (1994) when the stock prices follow the Geometric Levy process. For the extensions of Kolmogorov forward equation for Levy process, this paper uses adjoint operator in L² spaces. This paper obtains similar results of Dupire (1994) and Andersen and Andreasan (2001). However, our results can be applied to more general semi-martingale processes such as well-known VG (Variance Gamma) model and NIG (Normal Inverse Gaussian) model with diffusion processes. This paper also applies the approach to the case of stochastic time changed Levy process, which generates the stochastic volatility models.

We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. The dynamics of the default intensity are governed by the equation dλt=κ(ρ(Lt,t)−λt)dt+σλtdWt. The function ρ depends both on time t and accumulated loss Lt, providing sufficient freedom to calibrate the model to a generic distribution of loss. We develop a computationally efficient method for model calibration to the market of synthetic single tranche collateralized debt obligations (CDOs). The method is based on the Markovian projection technique which reduces the full model to a one-step Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function ρ can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX index tranches and apply it to the pricing of tranche options.

The interaction of two species can be considered to be either mutually competitive, the “co‐predator” case, or dominated by one species, the “predator‐prey” model. The study of stochastic versions of these models has been impeded by the occurrence of nonlinear expressions in the characterization of the resulting probability structure. Generating function methods are used to study the bivariate and conditional probability distributions and their moments. The results are illustrated using a Laplace transform technique for obtaining numerical values. These methods can be used in the development of ecological prediction techniques and can be extended easily to the bivariate logistic model.

The time delay would occurs when the measurements of multiple unmanned aerial vehicles (UAVs) are transmitted to the date processing center during cooperative target localization. This problem is often named as the out-of-sequence measurement (OOSM) problem. This paper aims to present a nonlinear filtering based on solving the Fokker–Planck equation to address the issue of OOSM.

According to the arrival time of measurement, the proposed nonlinear filtering can be divided into two parts. The non-delay measurement would be fused in the first part, in which the Fokker–Planck equation is utilized to propagate the conditional probability density function in the forward form. The time delay measurement is fused in the second part, in which the Fokker–Planck is used in the backward form approximately. The Bayes formula is applied in both parts during the measurement update.

Under the Bayesian filtering framework, this nonlinear filtering is not only suitable for the Gaussian noise assumption but also for the non-Gaussian noise assumption. The nonlinear filtering is applied to the cooperative target localization problem. Simulation results show that the proposed filtering algorithm is superior to the previous Y algorithm.

In this paper, the research shows that a better performance can be obtained by fusing multiple UAV measurements and treating time delay in measurement with the proposed algorithm.

In this paper, the OOSM problem is settled based on solving the Fokker–Planck equation. Generally, the Fokker–Planck equation can be used to predict the probability density forward in time. However, to associate the current state with the state related to OOSM, it would be used to propagate the probability density backward either.

The purpose of this paper is to discuss the application of the Haar wavelets for solving linear and nonlinear Fokker-Planck equations with appropriate initial and boundary conditions.

Haar wavelet approach converts the problems into a system of linear algebraic equations and the obtained system is solved by Gauss-elimination method.

The accuracy of the proposed scheme is demonstrated on three test examples. The numerical solutions prove that the proposed method is reliable and yields compatible results with the exact solutions. The scheme provides better results than the schemes [9, 14].

The developed scheme is a new scheme for Fokker-Planck equations. The scheme based on Haar wavelets is expended for nonlinear partial differential equations with variable coefficients.

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

The purpose of this paper is to address a novel method for solving parabolic partial differential equations (PDEs) in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method of Lines (ITMOL), is inspired in the Transversal Method of Lines (TMOL), with strong insight from the method of separation of variables.

The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic PDE for the evaluation of the time derivative. As will be demonstrated later, this key step is responsible for improving the accuracy of ITMOL over its predecessor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense.

In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent.

Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.

The purpose of this paper is to present an algorithm based on operational Tau method (OTM) for solving fractional Fokker‐Planck equation (FFPE) with space‐ and time‐fractional derivatives. Fokker‐Planck equation with positive integer order is also considered.

The proposed algorithm converts the desired FFPE to a set of algebraic equations using orthogonal polynomials as basis functions. The paper states some concepts, properties and advantages of proposed algorithm and its applications for solving FFPE.

Some illustrative numerical experiments including linear and nonlinear FFPE are given and some comparisons are made between OTM and variational iteration method, Adomian decomposition method and homotpy perturbation method.

Results demonstrate some capabilities of the proposed algorithm such as the simplicity, the accuracy and the convergency. Also, this is the first presentation of this algorithm for FFPE.

Technological advancements and growing complexity of many real-time systems, namely, communication, transportation, defense systems, etc., necessitate the importance to adopt a well-planned repair process such as phase type quasi-renewal process contributing to an improved system performance. Further, in an attempt to boost the role of maintenance as a financial benefactor, repairman’s multiple vacation policy is incorporated. Also, the significance of the degree of repair is illustrated while indicating the suitability of the matrix-analytic approach via the phase type quasi-renewal operating/repair times in reliability. The paper aims to discuss these issues.

The optimal replacement policy is obtained by employing the matrix-analytic method and minimum average cost rate.

The considered models make a significant contribution towards establishing that the matrix-analytic method, using the phase type quasi-renewal process, aids in reducing the computations and also fills the gap in the literature in the study of deteriorating systems. Availability and rate of occurrence of failures are evaluated in transient and steady-state regime.

This model differs from the existing models, in that, a repairman’s multiple vacation, delayed repair time and representation of the failure occurrence by a mixed Poisson process have been incorporated into the analysis. Also, time-dependent case and N-policy have been adopted to explore the optimality issues using phase type quasi-renewal process analytically. The numerical illustrations warrant that the maintenance policy proposed in this paper produces a considerably lower cost.