In this paper we establish a version of the Feynman-Kac formula for the stochastic heat equation with a multiplicative fractional Brownian sheet. We prove the smoothness of the density of the solution, and the Hölder regularity in the space and time variables.

In this article, we study a d-dimensional stochastic nonlinear heat equation (SNLH) with quadratic nonlinearity, forced by fractional space-time white noise: ∂ t u − ∆u = ρ 2 +Ḃ , ∈ [0, T ] x R d 0 φ. Two types of regimes are exhibited, depending on the ranges Hurst index H (H ..., d) (0, 1) d+1. particular, show that local well-posedness resulting from Da Prato-Debussche trick, is easily obtai...

Abstract Heat transfer in skin tissue is an area of interest for medical sciences. In this paper we intend to study fractional bioheat equation for heat transfer in skin tissue with constant and sinusoidal heat flux condition on skin surface. Numerical solutions are obtained by implicit finite difference method. We study the effect of anomalous diffusion in skin tissue and compare it with norma...

Fractional calculus is the generalization of integer-order calculus to rational order. This subject has at least three hundred years of history. However, it was traditionally regarded as a pure mathematical field and lacked real world applications for a very long time. In recent decades, fractional calculus has re-attracted the attention of scientists and engineers. For example, many researcher...

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in [0,1]×R , with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in [8] in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of [8], which is base...

A new mathematical model of time fractional order heat equation and fractional order boundary condition have been constructed in the context of the generalized theory of thermo piezoelasticity. The governing equations have been applied to a semi infinite piezoelectric slab. The Laplace transform technique is used to remove the time-dependent terms in the governing differential equations and the...