Let s and z be complex variables, (s) the Gamma function, and (s) ν = (s+ν) (s) for any complex ν the generalized Pochhammer symbol. The principal aim of the paper is to investigate the functionwhere α, β, γ ∈ C; Re(α) > 0, Re(β) > 0, Re(γ ) > 0 and q ∈ (0, 1) ∪ N . This is a generalization of the exponential function exp(z), the confluent hypergeometric function Φ(γ, α; z), the Mittag-Leffler function E α (z), the Wiman's function E α,β (z) and the function E γ α,β (z) defined by Prabhakar. For the function E γ,q α,β (z) its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Whittaker transforms, generalised hypergeometric series form, Mellin-Barnes integral representation with their several special cases are obtained and its relationship with Laguerre polynomials, Fox H -function and Wright hypergeometric function is also established.

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order and Riesz–Feller fractional derivative of order respectively. We obtain solution in terms of Fox’s H-function with some special cases, by using Fourier–Laplace transforms.

This paper is devoted to the study of a Wright-type hypergeometric function (Virchenko, Kalla and Al-Zamel in Integral Transforms Spec. Funct. 12(1):89-100, 2001) by using a Riemann-Liouville type fractional integral, a differential operator and Lebesgue measurable real or complex-valued functions. The results obtained are useful in the theory of special functions where the Wright function occurs naturally. MSC: 33C20; 33E20; 26A33; 26A99

This paper is devoted to the study of a generalized Mittag-Leffler function operator introduced by Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007

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This paper deals with the study of heat transfer and thermal damage in triple layer skin tissue using fractional bioheat model. Here, we consider three types of heating viz. sinusoidal heat flux, constant temperature and constant heat flux heating on skin surface. An implicit finite difference scheme is obtained by approximating fractional time derivative by quadrature formula and space derivative by central difference formula. The temperature profiles and thermal damage in the skin tissue are obtained to study the effect of fractional parameter [Formula: see text] on diffusion process for constant temperature and heat flux boundary heating on skin surface. A parametric study for sinusoidal heat flux at skin surface has also been made.

In recent times, graph based spectral clustering algorithms have received immense attention in many areas like, data mining, object recognition, image analysis and processing. The commonly used similarity measure in the clustering algorithms is the Gaussian kernel function which uses sensitive scaling parameter and when applied to the segmentation of noise contaminated images leads to unsatisfactory performance because of neglecting the spatial pixel information. The present work introduces a novel framework for spectral clustering which embodied local spatial information and fuzzy based similarity measure to tackle the above mentioned issues. In our approach, firstly we filter the noise components from original image by using the spatial and gray–level information. The similarity matrix is then constructed by employing a similarity measure which takes into account the fuzzy c-partition matrix and vectors of the cluster centers obtained by fuzzy c-means clustering algorithm. In the last step, spectral clustering technique is realized on derived similarity matrix to obtain the desired segmentation result. Experimental results on segmentation of synthetic and Berkeley benchmark images with noise demonstrates the effectiveness and robustness of the proposed method, giving it an edge over the clustering based segmentation method reported in the literature.