First-order linear homogeneous equations

Let us first investigate the properties of equations that involve only first-order derivatives, i.e., equations such that 

Let us lump the nxt(i= 1,2,. .., n) together into the n-column vector x and the af7 coefficients into the matrix AnXn which is not assumed to be symmetric. We now note the system of equations in the matrix form 

The matrix Ue U 1 is usually denoted eAt. In the common-dimension expansion form (Section 2.1.3), this matrix product reads 

The final step can be taken by defining the nxn matrix Q by its ith column 3lfx 0. 

The solution vector x therefore can be viewed as the weighted sum of n components eA with a corresponding weight matrix Q=L9I X0 (i = 1. n). This form is extremely useful to predict the rate of growth or decay for each individual component of the solution. 

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Since a and 3 are represented by 4 x 4 matrices, the wave function / must also be a four-component function and the Dirac wave equation (3.9) is actually equivalent to four simultaneous first-order partial differential equations which are linear and homogeneous in the four components of P. According to the Pauli spin theory, introduced in the previous chapter, the spin of the electron requires the wave function to have only two components. We shall see in the next section that the wave equation (3.9) actually has two solutions corresponding to states of positive energy, and two corresponding to states of negative energy. The two solutions in each case correspond to the spin components. 

The singularity of the system (0, 0), is an unstable stationary point for e > 0. Apparently, equations (3.71), (3.72) can be immediately solved, being the system of homogeneous first-order linear equations. The solutions have... 

Equations (2) and (3) represent two coupled, first order, linear, non-homogenous differential equations of the form dxidt + a(> x =. For radical pairs in liquid solution x and f(x) are... 

For a first order reaction (-r ) = kC, and Equation 8-147 is then linear, has constant coefficients, and is homogeneous. The solution of Equation 8-147 subject to the boundary conditions of Danckwerts and Wehner and Wilhelm [23] for species A gives... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

The preceding approach applies to all linear systems that is, those involving mechanisms in which only first-order or pseudo-first-order homogeneous reactions are coupled with the heterogeneous electron transfer steps. As seen, for example, in Section 2.2.5, it also applies to higher-order systems, involving second-order reactions, when they obey pure kinetic conditions (i.e., when the kinetic dimensionless parameters are large). If this is not the case, nonlinear partial derivative equations of the type... 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

Consider the system to possess a specific RTD, E(f), and that the reactor is fed with a homogeneous, perfectly mixed feed stream. If a first-order reaction takes place within this reactor the system will be described by linear equations. In this case, the reaction kinetics and the system residence time distribution totally define the conversion of reactant which would be achieved in that system. In other words, any reactor system possessing that particular RTD under consideration would give the same feed conversion... 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

We need the tangential components of E and H (namely Ex, Ey, Hx, and Hy) as explicit variables, so we want to get rid of Hz and Ez this is done by solving Eqs. (2.15.14) and (2.15.15) explicitly and simultaneously for Ez and Hz in terms of Ex, Ey, Hx, and Hy. These expressions for Ez and Hz are then substituted into the remaining four differential equations, to produce four linear homogeneous first-order differential equations in the four tangential field variables Ex, Ey, Hx, Hz. For convenience, we define a 4 x 1 generalized field vector dr. 

As our first example, we consider the case of a first-order homogeneous reaction A -> B in a laminar flow tubular reactor for which the global equation is linear in c (i.e. r( c)) — (c)) and is therefore completely closed. To obtain the range of convergence of the two-mode model, we need to consider only the local equation. In this specific case, the reduced model equations to all orders of p are then given by... 

It cun be shown that the most general solution of a coupled set of linear, homogeneous first-order equations, represented by Eq. (III.6A.3), has the form... 

The radiation balance of a layer with the thickness d having an infinitely large surface, irradiated homogeneously from one side with exciting radiation, is given by the solution of four coupled linear first-order differential equations (Eqs. 3.5-1...4). This is a boundary value problem, with the definitions given in Fig. 3.5-2. We are discussing... 

Remember 2.1 The general solution to nonhomogeneous linear first-order differential equations can be obtained as the product of function to be determined and the solution to the homogeneous equation. 

The kinetic equation for homogeneous systems is given by Eq. (7.47). The evolution equation for the zeroth-order moment of the NDF is null, which is due to the fact that the collision integral does not change the number of particles, or, more explicitly, f Cdf = 0. If the rate of change of the particle velocity (i.e. particle acceleration) is a linear function of the particle velocity (i.e. f = a + b ), then the evolution equation for the first-order moments are... 

Generalized-function formulations of GPT for homogeneous systems are the source of sensitivity functions for different integral parameters Equation (189) for reactivity worths, and Eq. (162) for ratios of linear and bilinear functionals. The first-order perturbation theory expression for reactivity [Eq. (132)] can also be used for sensitivity studies. 

This non-linear homogeneous differential equation of first order cannot be solved analytically. Therefore a numerical approach is used. A simple integration scheme is... 

The problem is separable for a bare homogeneous reactor. However, only the case of a step input of reactivity, i.e., the case of a constant value of p, is easily solved. In this case, the kinetic equations are readily reduced to a second order (for the case of one delayed neutron group) homogeneous linear differential equation with constant coefficients. For an input of positive reactivity two solutions arise, of the form and where o>i > 0 and 0)2 < 0. The first solution controls the persisting exponential rise of the flux, where it is recalled that T = l/o>i is the reactor period, and the second solution which rapidly becomes small is called the transient solution. 

The time dependent behavior of the flux is determined by solving a system of six linear non-homogeneous differential equations of the first order, with constant coefficients, involving Ty(t) and Hiv t). When the initial conditions are fulfilled the expression for the flux takes the form... 

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. 

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