Nth order differential equations

The above can be generalized to an Nth-order derivative with respect to time. In going ffom the time domain into the Laplace domain, d xjd is replaced by 5". Therefore an Nth-order differential equation becomes an Nth-order algebraic equation. 

We consider the solutions of the Nth order differential equations of the form ... 

If Eq. (2.114) is the characteristic equation for a real physical system, the coefficients o and a must be real numbers. These are the coefficients that multiply the derivatives in the Nth-order differential equation. So they cannot be imaginary. 

The same procedure presented above for second order differential equations can be extended to an nth order differential equation and to coupled nth order differential equations, as the reader will see in the homework problems. 

T.lj. Convert the following nth order differential equation to the standard format of Eq. 7.1 ... 

Often, an nth-order differential equation is placed in state-variable form. This is a set of n first-order equations. When deriving equations via material and energy balances, this state-variable form arises naturally. By developing the equations in state-variable form, we must solve the simultaneous sets of linear first-order differential equations. The state variable form is... 

Another important property of the transfer function is that the order of the denominator polynomial (in s) is the same as the order of the equivalent differential equation. A general linear nth-order differential equation has the form... 

In certain problems it may be necessary to locate all the roots of the equation, including the complex roots. This is the case in finding the zeros and poles of transfer functions in process control applications and in formulating the analytical solution of linear nth-order differential equations. On the other hand, different problems may require the location of only one of the roots. For example, in the solution of the equation of state, the positive real root is the one of interest. In any case, the physical constraints of the problem may dictate the feasible region of search where only a subset of the total number of roots may be indicated. In addition, the physical characteristics of ihe problem may provide an approximate value of the desired root. 

To obtain a unique solution of an nth-order differential equation or of a set of n simultaneous first-order differential equations, it is necessary to specify n values of the dependent variables (or their derivatives) at specific values of the independent variable. 

An nth order differential equation can in a similar manner be written in terms of an equivalent set of n first order differential equations. In the above set of equations, each equation has only one derivative term. While many coupled sets of equations are of this form, other sets of equations may have mixed derivative... 

Note that this is a second-order differential equation since the ancilla is chosen to be a two-level system. For an N-level ancilla, the equation would be of Nth order and read... 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol)" per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is... 

The basic differential equation for mass transfer accompanied by an nth order chemical reaction in a spherical particle is obtained by taking a material balance over a spherical shell of inner radius r and outer radius r + Sr, as shown in Figure 10.12. 

Whichever the type, a differential equation is said to be of nth order if it involves derivatives of order n but no higher. The equation in the first example is of first order and that in the second example of second order. The degree of a differential equation is the power to which the derivative of the highest order is raised after the equation has been cleared of fractions and radicals in the dependent variable and its derivatives. 

Figure 3.6 Numerical solution of ordinary differential equations sketch of the four steps of the Runge-Kutta method to the order four giving the n+1 th estimate y(n+1) from the nth estimate y ". ...

Equations (15)—(17) represent an ordinary differential equation and a boundary-value problem. For a suitable solution, dimensionless equations can be derived from Eqs. (15)—(17) by letting X = x/L, C = dc. For an isothermal reaction, that is nth-order and irreversible, in planar geometry we obtain... 

Derive the partial differential equation for unsteady-state unidirectional diffusion accompanied by an nth-order chemical reaction (rate constant k) ... 

Vanrolleghem 2006), worked on a differential equation derived from the original theory of crystallization described by Avrami (1939 1940 1941). The main hypothesis of this work is that crystallization could be interpreted as an nth order reaction and melting as a first order reaction,... 

Differential Data Analysis As indicated above, the rates can be obtained either directly from differential CSTR data or by differentiation of integral data. A common way of evaluating the kinetic parameters is by rearrangement of the rate equation, to make it linear in parameters (or some transformation of parameters) where possible. For instance, using the simple nth-order reaction in Eq. (7-165) as an example, taking the natural logarithm of both sides of the equation results in a linear relationship Between the variables In r, 1/T, and In C ... 

An nth-order chemical reaction with one reactant obeys the differential equation... 

What is the complementary solution, and what is the particular solution for (a) an nth-order linear differential equation, and (b) a 2 x 2 system of linear differential equations What do these solutions mean What factors determine them ... 

Consider a simple processing system with a single input and a single output (Figure 9.1a). The dynamic behavior of the process is described by an nth-order linear (or linearized nonlinear) differential equation ... 

It is not difficult to show that the complete integral of a differential equation of the nth order, contains n, and no more than n, arbitrary constants. As the reader acquires experience in the representation of natural processes by means of differential equations, he will find that the integration must provide a sufficient number of undetermined constants to define the initial conditions of the natural process symbolized by the differential equation. The complete solution must provide so many particular solutions (containing no undetermined constants) as there are definite conditions involved in the problem. For instance, equation (5), page 375, is of the third order, and the complete solution, equation (9), requires three initial conditions, g, s0, v0 to be determined. Similarly, the solution of equation (4), page 375, requires two initial conditions, m and 6, in order to fix the line. 

As a general rule it is more difficult to solve differential equations of higher orders than the first. Of these, the linear equation is the most important. A linear equation of the nth order is one in which the dependent variable and its n derivatives are all of the first degree and are not multiplied together. If a higher power appears the equation is not linear, and its solution is, in general, more difficult to find. The typical form is... 

When the coefficient of the differential equation is an nth-order polynomial, the n + 2 Stokes lines run radially in the asymptotic region. There are thus n + 2 unknown Stokes constants, but only three independent conditions are obtained from the singlevaluedness as demonstrated for the Airy function. 

Nonconstant density, 136 Nonhomogeneous equation, 117 Nonideal mixtures, 82 Nonisothermal pellet, 393 Nonlinear differential equation, 119 nth-order... 

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