Solution of a differential equation

In what follows it is supposed that w = u is a solution of a differential equation, say... 

A rigorous definition of stability of a difference scheme will be formulated in the next section. The improvement of the approximation order for a difference scheme on a solution of a differential equation will be of great importance since the scientists wish the order to be as high as possible. 

In the final analysis, of greatest importance from the viewpoint of numerical analysis is the design of algorithms permitting one to obtain a solution of a differential equation on a computer with a prescribed accuracy in a finite number of operations. The user can encounter in this connection the question of the quality of an algorithm, that is, the manner in which the accuracy of the algorithm depends on... 

If the dependent variable y(jt) and all of its derivatives occur in the first degree and do not appear as products, the equation is said to be linear. In effect, the solution of a differential equation of order n necessitates n integrations, each of which involves an arbitrary constant. However, in some cases one or more of these constants may be assigned specific values. The results, which are also solutions of the differential equation, are referred to as particular solutions. The general solution, however, includes all of the n constants of integration, whose evaluation requires additional information associated with the application. 

A spinning electron also has a spin quantum number that is expressed as 1/2 in units of ti. However, that quantum number does not arise from the solution of a differential equation in Schrodinger s solution of the hydrogen atom problem. It arises because, like other fundamental particles, the electron has an intrinsic spin that is half integer in units of ti, the quantum of angular momentum. As a result, four quantum numbers are required to completely specify the state of the electron in an atom. The Pauli Exclusion Principle states that no two electrons in the same atom can have identical sets of four quantum numbers. We will illustrate this principle later. 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

On the one hand, a property called cooperativity will be used. This property must hold upon the dynamics of the observation error associated to (19). The cooperative system theory enables to compare several solutions of a differential equation. More particularly, if a considered system = /(C, t) is cooperative, then it is possible to show that given two different initial conditions defined term by term as i(O) < 2(0) then, solutions to this system will be obtained in such a way that i(t) < 2(t), where 1 and 2 are the solutions of the differential equations system with the initial conditions (0) and 2(0), respectively. This is exactly the same result established previously in the case of simple mono-biomass/mono-substrate systems. With regard to this property the following lemma is recalled. 

The streamline function is the solution of a differential equation, the details of which we will not pursue. We will have more to say about flow streamlines in our discussion of viscosity in Chapter 4. For the present, however, it is sufficient to note that an important part of... 

It should be noted that, in most chemical situations, we rarely need the general solution of a differential equation associated with a particular property, because one (or more) boundary conditions will almost invariably be defined by the problem at hand and must be obeyed. For example ... 

The solution methodology of the determinants is similar to that of the well-known Thomas algorithm used for the numerical solution of a differential equation with the finite-difference method [50]. An essential difference from the Thomas algorithm is that the first step ofthe algorithm here is a so-called backward process. This means that the calculation of T starts from the last sublayer, that is, from the Mth sublayer ofthe determinant and it is continued down to the 1st sublayer. Thus, the value of Ti is obtained directly, in the fist calculation step. Then, applying the known value of Ti, the value of Pi can be obtained by means of the fist boundary condition at X= 0, namely ... 

The solution to the first problem is limited by the increase in time or the computer capacity available to solve more complete or more advanced equations. The second problem is even more difficult to acknowledge. It may be due to error accumulation through the nonlinear domain. The numerical solution of a differential equation is based on the approximation of time and, in the case of PDEs, space partial derivatives, by finite-difference equivalents. 

New functions are sometimes defined as a solution to differential equation, and simply named after the differential equation itself. It is the purview of the mathematician to understand the properties of these functions so that they can be used confidently in numerous other applications. The Bessel function is of this kind, the solution of a differential equation that occurs in many applications of engineering and physics, including heat transfer. 

We allow ourselves a short digression here, in order to make a special point. There are two ways of presenting an error in a numerical solution of a differential equation. The usual way is to refer to the error in the quantity computed at each new time interval that is, the difference between the numerical approximation and the exact solution (if it is known). Another way is to compute, for each calculated value, the time at which that value is exact, and to express the error as a time shift, the difference between the calculated time and the exact time at that iteration number. It is called a time shift because in many kinds of simulations dealt with in this book, time itself does not enter the equations and, once a simulated sequence of values has become shifted along in time, that shift is permanent. Putting this another way, there is no clock inherent in the method. It will be seen (Chap. 8) that in fact, in... 

It follows also from Eq. (1-3) that there exist electron states having discrete or definite values for energy (or, states with discrete values for any other observable). This can be proved by construction. Since any measured quantity must be real, Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics that it is possible to construct eigenstates of any Hermitian operator. However, for the Hamiltonian operator, which is a Hermitian operator, eigenstates are obtained as solutions of a differential equation, the time-independent Schroedinger equation. 

Whenever you obtain the solution of a differential equation, you should check it as many ways as you can. In this case, we have three ways ... 

You will recall from algebra that a number is a solution of an algebraic equation if it satisfies the equation. For example, 2 is a solution of the equation. v -8 = 0 because the subslitutiou of 2 for jr yields identically zero. Likewise, a function is a solution of a differential equation if that function satisfies the differential equation, In other words,.a solution function yields identity when substituted into the differential equation. For example, it can be shown by direct substitution that the function is a solution of / - 4y - 0 O ig. 2-72). 

Since the a and n MOs are mathematical functions obtained in principle as solutions of a differential equation, describing the motion of a single electron in the field provided by nuclei and all remaining electrons, we... 

For example, the free particle wavefunction (2.200), a solution of a differential equation of the second-order, is also characterized by two coefficients, and we may choose 5 = 0 to describe a particle going in the positive x direction or ff = 0 to describe a particle going in the opposite direction. The other coefficient can be chosen to express normalization as was done in Eq. (2.82). 

Rule 4 Special conditions holding for the solutions of a differential equation should be equivalently valid for the solutions of the finite-difference scheme. 

This example illustrates that the step size, h, plays a crucial role in the convergence of the numerical solution of a differential equation. Depending on the equation to be solved and the method of discretization, h can be adjusted to obtain convergence of a numerical solution. 

The solution of a differential equation is a function whose derivative or derivatives satisfy the differential equation. 

A differential equation is an equation that contains one or more derivations of an unknown function. The solution of a differential equation is the unknown function, not a set of constant values of an unknown variable as is the case with an algebraic equation. Our first examples of differential equations are equations of motion, obtained from Newton s second law of motion. These equations are used to determine the time dependence of the position and velocity of particles. The position of a particle is given by the position vector r with Cartesian components x, y, and z. The velocity v of a particle is the rate of change of its position vector. 

The statement DSolve is used to carry out a symbolic solution of a differential equation. We illustrate this with an example. 

We annotate that the method we have shown is to find an approximate solution of a differential equation near a fixed poinf via Taylor expansion. Here, as we could solve all terms of the expansion, we have found an analytical solution of the differential equation. 

To paraphrase the Greek philosopher Heraclitus, The only thing constant is change. Differential equations describe mathematically how things in the physical world change— both in time and in space. While the solution of an algebraic equation is usually a number, the solution of a differential equation gives SL function. 

Further special rules are found to be appropriate for the treatment of problems about the transitions of systems from one energy state to another. Suppose the wave equation has solutions corresponding to two permitted energy levels, and E. If there are two possible solutions of a differential equation, it is easily seen that the sum of the two, or indeed any linear combination of them, is also a solution. Thus if and are solutions, then... 

The solution of a differential equation consists of a special trend univocally determined by the differential equation itself and the initial conditions. Figure 2.2 shows an ill-conditioned equation formulation small perturbations in the initial conditions or small deviations from the solution lead to completely different trends. 

The general solution of a differential equation of nth order usually has n arbitrary constants. To fix these constants, we may have boundary conditions, which are conditions that specify the value of y or various of its derivatives at a point or points. Thus, if y represents the displacement of a vibrating string held fixed at two points, we knowy must be zero at these points. 

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