Linear differential equation Exact

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... 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

R.I. Jennrich and P.B. Right, Fitting systems of linear differential equations using computer generated exact derivatives, Technometrics,... 

For a CSTR the stationary-state relationship is given by the solution of an algebraic equation for the reaction-diffusion system we still have a (non-linear) differential equation, albeit ordinary rather than partial as in eqn (9.14). The stationary-state profile can be determined by standard numerical methods once the two parameters D and / have been specified. Figure 9.3 shows two typical profiles for two different values of )(0.1157 and 0.0633) with / = 0.04. In the upper profile, the stationary-state reactant concentration is close to unity across the whole reaction zone, reflecting only low extents of reaction. The profile has a minimum exactly at the centre of the reaction zone p = 0 and is symmetric about this central line. This symmetry with the central minimum is a feature of all the profiles computed for the class A geometries with these symmetric boundary conditions. With the lower diffusion coefficient, D = 0.0633, much greater extents of conversion—in excess of 50 per cent—are possible in the stationary state. 

From a conceptual point of view, nothing new is needed to further extend the above approach. For instance, the one-box model with two variables shown in Fig. 21.7 can be combined with the two-box (epilimnion/ hypolimnion) model. This results in four coupled differential equations. Even if the equations are linear, it is fairly complicated to solve them analytically. Computers can deal more efficiently with such problems, thus we refrain from adding another example. But we should always remember that independently from how many equations we couple, the solutions of linear models always consist of the sum of a number of exponential terms which have exactly one steady-state, although it may be at infinity. In Section 21.4 we will discuss the general structure of linear differential equations. 

For the special case of non relativistic Hydrogen, the multiphoton transition rate can be obtained exactly using methods based on Green function techniques, which avoid summations over intermediate states. This approach was introduced in order to treat time independent problems, and later extended to time dependent ones [2]. In the Green function method, the evaluation of the infinite sums over intermediate states is reduced to the solution of a linear differential equation. For systems other than Hydrogen, this method can also be used, but the associated differential equation has to be integrated numerically. The two-photon transition rate can also be evaluated exactly by performing explicitly the summation over the intermediate states. 

Samalam [43] modeled the convective heat transfer in water flowing through microchannels etched in the back of silicon wafers. The problem was reduced to a quasi-two dimensional non-linear differential equation under certain reasonably simplified and physically justifiable conditions, and was solved exactly. The optimum channel dimensions (width and spacing) were obtained analytically for a low thermal resistance. The calculations show that optimizing the channel dimensions for low aspect ratio channels is much more important than for large aspect ratios. However, a crucial approximation that the fluid thermophysical properties are independent of temperature was made, which could be a source of considerable error, especially in microchannels with heat transfer. 

A differential equation contains one or more derivatives of an unknown function, and solving a differential equation means finding what that function is. One important class of differential equations consists of classical equations of motion, which come from Newton s second law of motion. We will discuss the solution of several kinds of differential equations, including linear differential equations, in which the unknown function and its derivatives enter only to the first power, and exact differential equations, which can be solved by a line integration. We will also introduce partial differential equations, in which partial derivatives occur and in which there are two or more independent variables. We will also discuss the solution of differential equations by use of Laplace transformations. Some differential equations can be solved either symbolically or numerically using Mathematica. 

This is a linear differential equation with constant coefficients that can be solved by conventional techniques. In this case, the coefficients are A2 and A1 Ni. This equation has exactly the same form as that which results when describing series reactions, and its solution was presented in Section 2.4. After assuming a solution of the form... 

Modern research has concentrated on elucidating the nature of L. If one has an equation representative of Eq. (3), questions abound concerning L, questions more subtle than the obvious what is it Equation (3) is irreversible in time, while any exact equation for A (t) must be reversible. This dilemma could not be resolved with irreversible thermodynamics. Also, consider that Eq. (3) may represent a set of coupled, linear differential equations. Suppose a certain set of variables A must be taken together to obtain proper dynamics. Then, if one tried to apply the principles of irreversible thermodynamics with a set smaller than the true set, an incorrect dynamical law would result no matter what was used for L. How was one to know the correct set of A Finally, the obvious question does occur Given that it is reasonable to write an irreversible equation and given that one has a proper set of variables, how does one find an expression for L ... 

In recent decades the thinking of physicists has largely been dominated by attempts to describe systems in terms of linear differential equations and their solutions. Deviations fi om their harmonic behaviour, which lead to non-linear terms in the differential equations, have been treated as perturbations by introducing interactions between the quasi-particles, correspond to the harmonic solutions (electron-electron and electron-phonon collisions, etc.). The idea of the soliton concept is to solve the non-linear differential equations, not by numerical approximations but analytically and to associate new quasi-particles wifli exact solutions, the solitons. 

Here b is the radius of curvature at the particle apex, where the two principal curvatures are equal (e.g., the bottom of the bubble in Figure 4.11a). Unfortunately, Equation 4.107, along with Equation 4.114, has no closed analytical solution. The meniscus shape can be exactly determined by numerical integration of Equation 4.110. Alternatively, various approximate expressions are available [199,209,210]. For example, if the meniscus slope is small, z linear differential equation of Bessel type, whose solution reads... 

The above expression is a differential equation, since the value j/K (named as the source function) depends on the intensity of the radiation at each point. A solution to this expression can be obtained only by approximation. In the exact solution, the equation requires the division of the radiation field into a large number n of linear differential equations. The detailed solution has been presented by S. Chandrasekhar (2). Such a rigorous solution is practically never used for the calculation of isotropic scattering in the thin layer. 

The examples of computational accuracy so far have been for linear differential equations. This is reasonable since for such equations the exact solution of such linear equations is known. It would also be instructive to explore similar results for nonlinear equations. In general, nonlinear differential equations do not have closed form solutions so one must extrapolate to such equations. There are, however, a small set of nonlinear differential equations for which an exact solution can be obtained and one such equation is the second order equation ... 

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. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. Completeness means that any arbitrary function can be expressed exactly as a linear combination of these functions. Mathematically, completeness can be expressed as... 

With K being a constant, an exact solution to the linear ordinary differential equation gives the velocity profile as... 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

The laminar flow assumption eliminates the non-linear term in the partial differential equations system (3.3), thus significantly reducing the computational cost. In addition, the present formulation often admits an exact solution. For example, in the case of an incompressible 2D laminar flow between two motionless parallel plates (i.e. planar SOFC configuration of Figure 3.1), Equation (3.29) reduces to ... 

For purely intramolecular equilibria the system of differential equations (72) is already linear and the procedure described above transforms the system into the form of equation (104) exactly without any approximations. For such equilibria further simplification of equations (72) and (104) is possible by deleting all those quantities which refer to empty sets of nuclei. 

For the case of the linear spring given in equation (7.45), equation (7.52) is a linear, ordinary differential equation that can be solved exactly. If the initial condition is a qui-... 

The formulas derived above, despite their cumbersome look, are very practical. Indeed, they present the nonlinear initial susceptibilities of a superparamagnetic particulate medium as analytical expressions of arbitrary accuracy. Another remarkable feature of the formulas of Section III.B.6 is that with respect to the frequency behavior they give the exact structure of the susceptibilities and demonstrate that those dependencies are quite simple. This makes our formulas a handy tool for analytical studies. Yet they are more convenient for numerical work because with their use the difficult and time-consuming procedure of solving the differential equations is replaced by a plain summation of certain power series. For example, if to employ Eqs. (4.194)-(4.200), a computer code that fits simultaneously experimental data on linear and a reasonable set of nonlinear susceptibilities (say, the 3th and the 5th) taking into account the particle polydispersity of any kind (easy-axes directions, activation volume, anisotropy constants) becomes a very fast procedure. 

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