a b s t r a c tIn this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.
ABSTRACT:This study provides an analysis of the convergence of the Haar wavelet-based method for solving twodimensional boundary value problems. The convergence analysis shows that the approximation method is of order 2. The analytical results are validated via two numerical examples.
This research demonstrates the log-convexity and log-concavity of the modified Bessel function of the first kind and the related functions. The method of coefficient is used to verify such properties. One of our results contradicts the conjecture proposed by Neumann in 2007 which states that modified Bessel function of the first kind I ν is log-concave in (0, ∞) given ν > 0. The log-concavity holds true in some bounded domain. The application of the other results in Kibble's bivariate gamma distribution is also demonstrated.
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