Exact differential equations

The Maxwell relations in thermodynamics are obtained by treating a thermodynamic relation as an exact differential equation. Exact differential equations are of the form... 

In the T-method one admits that one cannot solve the linear differential equation exactly, and inserts an error term tF (x), where t is an a priori undetermined coefficient, and P x) is a known orthogonal polynomial of order n. Thus, one writes... 

Zaitsev, V. F. and Polyanin, A. D., Handbook of Partial Differential Equations. Exact Solutions, MP Obrazovaniya, Moscow, 1996 [in Russian],... 

Orthogonal collocation - the substitution of the concentration profiles by polynomials that are forced to locally fulfil the differential equations exactly at certain points (discretization based on the zeroes of the polynomials). The partial differential equations are reduced to ordinary differential equations to be solved by standard methods. 

In other words, the function approximation methods find a solution by assuming a particular type of function, a trial (basis) function, over an element or over the whole domain, which can be polynomial, trigonometric functions, splines, etc. These functions contain unknown parameters that are determined by substituting the trial function into the differential equation and its boundary conditions. In the collocation method, the trial function is forced to satisfy the boundary conditions and to satisfy the differential equation exactly at some discrete points distributed over the range of the independent variable, i.e. the residual is zero at these collocation points. In contrast, in the finite element method, the trial functions are defined over an element, and the elements, are joined together to cover an entire domain. 

All of these quantities are state fiinctions, i.e. the differentials are exact, so each of the coefficients is a partial derivative. For example, from equation (A2.1.35) p = —while from equation (A2.1.36)... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

We have in mind trajectory calculations in which the time step At is large and therefore the computed trajectory is unlikely to be the exact solution. Let Xnum. t) be the numerical solution as opposed to the true solution Xexact t)- A plausible estimate of the errors in X um t) can be obtained by plugging it back into the differential equation. 

They represent the generalization of our earlier equations (11.12) and include an additional equation for the pressure, since this can no longer be assumed to take the constant value p throughout the pellet. The differential equation relating X to f is derived exactly as before and has the same form, except that p cannot be replaced by p In the present case. Thus we have... 

It is a property of linear, homogeneous differential equations, of which the Schroedinger equation is one. that a solution multiplied by a constant is a solution and a solution added to or subtracted from a solution is also a solution. If the solutions Pi and p2 in Eq. set (6-13) were exact molecular orbitals, id v would also be exact. Orbitals p[ and p2 are not exact molecular orbitals they are exact atomic orbitals therefore. j is not exact for the ethylene molecule. 

A relation between the variables, involving no derivatives, is called a solution of the differential equation if this relation, when substituted in the equation, satisfies the equation. A solution of an ordinaiy differential equation which includes the maximum possible number of arbitrary constants is called the general solution. The maximum number of arbitrai y constants is exactly equal to the order of the dif-... 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

Equations (B.15) are exactly the same as those derived by Holstein [1978], and the following discussion draws on that paper. The pair of equations (B.15) may be represented as a single second-order differential equation... 

That is. Equation (5.29) satisfies the differential equation. Equation (5.27), and the boundary conditions. Equation (5.28), so is the exact solution if... 

Solving the gas dynamics expressions of Kuhl et al. (1973) requires numerical integration of ordinary differential equations. Hence, the Kuhl et al. paper was soon followed by various papers in which Kuhl s numerical exact solution was approximated by analytical expressions. 

Following exactly the procedure applied in the earlier example, these differential equations are transformed into algebraic equations. 

While one is free to think of CA as being nothing more than formal idealizations of partial differential equations, their real power lies in the fact that they represent a large class of exactly computable models since everything is fundamentally discrete, one need never worry about truncations or the slow aciminidatiou of round-off error. Therefore, any dynamical properties observed to be true for such models take on the full strength of theorems [toff77a]. 

A difference between these two concepts can be illustrated in many ways. Consider, for example, a mathematical pendulum in this case the old concept of trajectories around a center holds. On the other hand, in the case of a wound clock at standstill, clearly it is immaterial whether the starting impulse is small or large (as long as it is sufficient for starting, the ultimate motion will be exactly the same). Electron tube circuits and other self-excited devices exhibit similar features their ultimate motion depends on the differential equation itself and not on the initial conditions. 

In equation (1.4), the infinitesimal change dZ is an exact differential. Later, we will describe the mathematical test and condition to determine if a differential is exact. 

By using relationships for an exact differential, equations that relate thermodynamic variables in useful ways can be derived. The following are examples. 

Since dS is an exact differential, equations for dS = 0 can be integrated. The integration yields a family of solution surfaces, S = S(.vi,... x ) = constant. Each solution surface contains a set of thermodynamic states for which the entropy is constant.hh... 

Application of the Maxwell relations equation (Al. 28) will show that this differential is exact. Integration leads to a family of surfaces... 

Within each solution surface are numerous subsets of points that also satisfy the differential equation bQ = dF = 0. These subsets are referred to as solution curves of the Pfaffian. The curve z — 0, y + y2 = 25.00 is one of the solution curves for our particular solution surface with radius = 5.00. Others would include x = 0, y2 + z2 — 25.00, and r — 0,. v2 + r2 = 25.00. Solution curves on the same solution surface can intersect. For example, our first two solution curves intersect at two points (5, 0, 0) and (-5, 0. 0). However, solution curves on one surface cannot be solution curves for another surface since the surfaces do not intersect. That two solution surfaces to an exact Pfaffian differential equation cannot intersect and that solution curves for one surface cannot be solution curves for another have important consequences as we see in our discussion of the Caratheodory formulation of the Second Law of Thermodynamics. 

Thus, exact or integrable Pfaffians lead to non-intersecting solution surfaces, which requires that solution curves that lie on different solution surfaces cannot intersect. For a given point p. there will be numerous other points in very close proximity to p that cannot be connected to p by a solution curve to the Pfaffian differential equation. No such condition exists for non-integrable Pfaffians, and, in general, one can construct a solution curve from one point to any other point in space. (However, the process might not be a trivial exercise.)... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

Population Density Response Surface. The algorithm method of characteristic is used to reduce the partial differential Equation (1) into a set pf coupled ordinary differential equations. Since T(n,t) is an exact differential, then... 

Stability of difference schemes with respect to coefficients. In solving some or other problems for a differential equation it may happen that coefficients of the equation are specified not exactly, but with some error because they may be determined by means of some computational algorithms or physical measurements, etc. Coefficients of a homogeneous difference scheme are functionals of coefficients of the relevant differential equation. An error in determining coefficients of a scheme may be caused by various... 

Equation (28) is the set of exact coupled differential equations that must be solved for the nuclear wave functions in the presence of the time-varying electric field. In the spirit of the Born-Oppenheimer approximation, the ENBO approximation assumes that the electronic wave functions can respond immediately to changes in the nuclear geometry and to changes in the electric field and that we can consequently ignore the coupling terms containing... 

---