In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...

in the present work the effect of cu-water nanofluid, as heat transfer fluid, on the performance of a parabolic solar collector was studied numerically. the temperature field, thermal efficiency, mean-outlet temperatures have been evaluated and compared for the conventional parabolic collectors and nanofluid based collectors. further, the effect of various parameters such as fluid velocity, vol...

We consider Kolmogorov operator $$-\nabla \cdot a \nabla + b $$ with measurable uniformly elliptic matrix and prove Gaussian lower upper bounds on its heat kernel under minimal assumptions the vector field divergence $$\mathrm{div\,}b$$ . More precisely, we prove: (1) bound, provided that $$\mathrm{div\,}b \ge 0$$ , is in class of form-bounded fields (containing e.g. $$L^d$$ weak class, as well...

We focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of It¨o type, in particular, parabolic equations. The main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.

In this paper difference methods to solve "fuzzy partial differential equations" (FPDE) such as fuzzy hyperbolic and fuzzy parabolic equations are considered. The existence of the solution and stability of the method are examined in detail. Finally examples are presented to show that the Hausdorff  distance between the exact solution and approximate solution tends to zero.

To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantit...

Stabilization of (ordinary) stochastic differential equations (SDEs) by noise has been studied extensively in the past few years, e.g., Arnold et al. [1], Has’minskii [6], Mao and Yuan [10], Pardoux and Wihstutz [11, 12], Scheutzow [15]. Recently, there are also many works focusing on such phenomena for stochastic partial differential equations (SPDEs), e.g., Kwiecinska [7] and Kwiecinska [9] d...

Recently M. Crandall and P. L. Lions [3] developed a very successful method for proving the existence of solutions of nonlinear second-order partial differential equations. Their method, called the theory of viscosity solutions, also applies to fully nonlinear equations (in which even the second order derivatives can enter in nonlinear fashion). Solutions produced by the viscosity method are gu...

We study a new mathematical model of locally nonequilibrium processes heat, mass, and momentum transfer taking into account the relaxation phenomena based on hyperbolic parabolic equations. also propose method aimed at getting priori Schauder-type estimates. The unique solvability problem with inner-boundary nonlocal conditions is established.