Partial differential equations inhomogeneous

Inhomogeneous second-order (partial) differential equations... 

Center manifold theory extends to many infinite-dimensional systems, like certain partial differential equations (PDFs). Center manifold reductions can be obtained locally or globally. For local center manifolds of parabolic PDFs see Vanderbauwhede and looss [78]. Dimension reductions via global center manifolds for spatially inhomogeneous planar media have been achieved by Jangle [27, 33] more details will be presented below. 

Coleman CJ, Tullock D, Phan-Thien N (1991) An effective boundary element method for inhomogeneous partial differential equations. Z Angew Math Phys 42 730-745 Coppola S, Grizzuiti N, Maffettone PL (2001) Microrheological modeling of flow-induced crystallization. Macromol 34 5030-5036... 

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

Great efforts are needed even in a laboratory to achieve a homogeneous spatial distribution of the concentrations, temperature and pressure of a system, even in a small volume (a few mm or cm ). Outside the confines of the laboratory, chemical processes always occur under spatially inhomogeneous conditions, where the spatial distribution of the concentrations and temperature is not uniform, and transport processes also have to be taken into account. Therefore, reaction kinetic simulations frequently include the solution of partial differential equations that describe the effect of chemical reactions, material diffusion, thermal diffusion, convection and possibly turbulence. In these partial differential equations, the term f defined on the right-hand side of Eq. (2.9) is the so-called chemical source term. In the remainder of the book, we deal mainly with the analysis of this chemical source term rather than the full system of model equations. 

Unfortunately, obtained equation could not be solved by means of Mathcad symbolic core directly . We definitely can differentiate it with respect to variable t and get linear inhomogeneous differential equation of second-order with respect to derivative. After that we can use the methodic of getting partial solution, given nearly in every handbook of differential equations. However it will mainly be a hand work, and not a computational calculation. Symbolic resources of Maple allow finding the solution of its equation directly and getting analytic expressimi for time-dependence of intermediate s concentration (Fig. 1-9) ... 

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