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The aim of this paper is to develop mathematical theory of the conventional and invariant schemes for the method of fundamental solutions used to solve potential problems in doubly-connected regions. Particularly, we prove that an approximate solution actually exists uniquely under some conditions, and that the error decays exponentially when the boundary data are analytic, and algebraically when they are not analytic but belong to some Sobolev spaces. Moreover, we present results of several numerical experiments in order to show the sharpness of our error estimate.

We present the convergence rates and the explicit error bounds of Hill’s method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is self-adjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin’s theorem for some real problems. Main theorems are proved using the Dunford integrals which project an vector to a specific eigenspace.

This paper discusses the potential application of Fuzzy set theory in the area of risk assessment using linguistic variables/values. The application has been illustrated through its use in an open cast Iron Ore-Mine. Yager’s methodology for Ordinal multiobjective decisions based on fuzzy sets has been chosen since the model only demands ordinal information of experts’ preferences and the importance of each individual factor. Moreover this model can be used by the practising Engineers who may not be having in-depth knowledge on fuzzy mathematics. Typical hazards associated with open cast mining have been used in this application.

In this paper, we propose a numerical method for one-dimensional Fredholm integral equations of the second kind by the IMT-type DE rules for numerical integration. We obtain our method by enhancing the DE-Nyström method by replacing the DE rule used for discretizing the integral operator with the IMT-type DE rules. It is free of the difficulty of parameter tuning, that is, the problem of choosing the mesh size of the DE rule for the given number of unknowns as in the DE-Nyström method. Numerical examples show that it is competitive with the DE-Nyström method.

This paper investigates a renewal risk model with stochastic return and Brownian perturbation, where the price process of the investment portfolio is described as a geometric Lévy process. When the claim sizes have a subexponential distribution, we derive the asymptotics for the finite-time ruin probability of the above risk model. The obtained result confirms that the asymptotics for the finite-time ruin probability of the risk model with heavy-tailed claim sizes are insensitive to the Brownian perturbation.

A model with time delay is considered for a predator-prey system. Here, a single species disperses between n patches of a heterogeneous environment with barriers between patches while a predator does not involve a barrier between patches. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are established for the global asymptotic stability of the positive equilibrium of the system.

In this paper, improved algorithms are proposed for preconditioned bi-Lanczos-type methods with residual norm minimization for the stable solution of systems of linear equations. In particular, preconditioned algorithms pertaining to the bi-conjugate gradient stabilized method (BiCGStab) and the generalized product-type method based on the BiCG (GPBiCG) have been improved. These algorithms are more stable compared to conventional alternatives. Further, a stopping criterion changeover is proposed for use with these improved algorithms. This results in higher accuracy (lower true relative error) compared to the case where no changeover is done. Numerical results confirm the improvements with respect to the preconditioned BiCGStab, the preconditioned GPBiCG, and stopping criterion changeover. These improvements could potentially be applied to other preconditioned algorithms based on bi-Lanczos-type methods.

A new implicit Runge-Kutta-Nyström method with variable coefficients is developed for solving the periodic initial value problem of the differential equationy″ = f(t,y). The proposed method, whose coefficients are functions of the frequency and the stepsize, integrates exactly the equation, if the solution is a periodic function with a single Fourier component and the frequency is known. On the other hand, the order of accuracy of the method is shown to be 4 for the case that an estimated frequency, instead of the exact one, is applied to evaluate the coefficients, as well as for that the solution is non-periodic.

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an $$\alpha$$ -stable process with $$\alpha \in (1,2)$$ . More precisely, we find an upper bound for the $$L^{\alpha -1}(\varOmega ,{\mathbb {P}})$$ distance between two solutions in terms of the $$L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)$$ distance of the coefficients for an appropriate measure $$\mu^{\alpha} _{x_0}$$ which characterizes symmetric stable laws and depends on the initial value of the stochastic differential equation. We obtain this result using the method introduced by Komatsu (Proc Jpn Acad Ser A Math Sci 58(8):353–356, 1982) which is used in the proof of uniqueness of solutions together with an upper bound for the transition density function of the solution of the stochastic differential equation obtained by Kulik (The parametrix method and the weak solution to an SDE driven by an $$\alpha$$ -stable noise. arXiv:1412.8732 , 2014).