Differential equations numerical solution, 29-31 Runge-Kutta

Equations in this form can sometimes be solved analytically. If this is not possible, a numerical solution may be accomplished. One very common numerical technique that is employed in practice to solve non-linear differential equations is the Runge-Kutta method. See Part IV, Chapter 21 for additional details. ... 

Brenan, K. E., and Petzold, L. R., The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods," UCRL-95905, preprint, Lawrence Livermore National Laboratories, Livermore, California (1987). 

Other transdermal systems give rates of release which are proportional to the square root of time. In order to model this behaviour it is possible to write a series of linear differential equations to describe transfer from the device and across the skin. However unlike the cases of first and zero order input, t1 2 input does not produce a simple analytical solution of the type given in equation (5). Plasma levels have therefore been calculated using a numerical approach and by solving the equations using the Runge-Kutta method. For GTN delivery, identical rate constants to... 

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 ". ...

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

Starting with an initial value of cA and given c,(t), Eq. (8-4) can be solved for cA(t + At). Once cA(t + At) is known, the solution process can be repeated to calculate cA(t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. As discussed in Sec. 3, more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Runge Kutta method, which involves the following calculations ... 

M is a singular Matrix. Zero entries on the main diagonal of this matrix identify the algebraic equations, and all other entries which have the value 1 represent the differential equation. The vector x describes the state of the system. As numeric tools for the solution of the DAE system, MATLAB with the solver odel5s was used. In this solver, a Runge Kutta procedure is coupled with a BDF procedure (Backward Difference Formula). An implicit numeric scheme is used by the solver. 

However because the rate law expressions for iron sulfide formation (equations 8 and lO) are non-linear the differential equations for H2S and the iron sulfides are not amenable to explicit solution Thus it is important to develop an equation for that can be incorporated in a numerical solution technique such as that of Runge-Kutta (15) Fortunately an appiropriate differential equation for can be developed firom charge balance considerations Here it is assumed that dissolved substances other than those listed in Table I are not affected by diagenesis If this is true, then a charge balance difference equation can be written (16) ... 

The Rimge-Kutta methods for numerical solution of the differential equation dy/dx = F(x, y) involve, in effect, the evaluation of the differential function at intermediate points between xi and Xj+i. The value of yi+ is obtained by appropriate summation of the intermediate terms in a single equation. The most widely used Runge-Kutta formula involves terms evaluated at X(, Xj + Ax/2 and X + Ax. The fourth-order Runge-Kutta equations for dy/dx = F(x, y) are... 

E(u,v) is the inner product of u and v. In conventional OCFE calculation methods, the exponent of Az in Eq. 10.112 is 6, hence the degree of convergence between the calculated and the true profiles is of the sixth degree with respect to the space increment. One expects the value of C I4 to be rather small in the type of problems dealt with here. The fourth-order Runge-Kutta method used in the OCFE algorithm discussed here introduces an error of the fifth order. Accordingly, we may anticipate that the numerical solutions of the system of partial differential equations of chromatography calculated by an OCFE method will be more accurate than those obtained with a finite difference method [48] or even with the controlled diffusion method [49,50]. 

The Runge-Kutta methods for numerical solution of the differential equation dy/dx = F(x, y) involve, in effect, the evaluation of the differential function at intermediate points between x and The value of i/I+i is obtained by... 

If a mathematical method for solving a differential equation cannot be found, numerical methods exist for generating numerical solutions to any desired degree of accuracy. Euler s method and the Runge-Kutta method were presented. 

And finally, the new values of the concentrations at the time r + Ar can be calculated using a numerical solution of this latter ordinary differential equation. In this case, Euler s method (Perry, Green, and Malone, 1984) is used due to its simplicity, although errors are proportional to AL Other method of high order, as Runge-Kutta (Perry, Green, and Malone, 1984), can be used if needed ... 

The differential equations are stiff that is, several processes are going on at the same time, but at widely differing rates. This is a common feature of chemical kinetic equations and makes the numerical solution of the differential equations difficult. A steady state is never reached, so the equations cannot be solved analytically. Traditional methods, such as the Euler method and the Runge-Kutta method, use a time step, which must be scaled to fit the fastest process that is occurring. This can lead to large number of iterations even for small time scales. Hence, the use of Stella to model this oscillatory reaction would lead to an impossible situation. 

In [211] the authors obtained a new embedded 4(3) pair explicit four-stage fourth-order Runge-Kutta-Nystrom (RKN) method to integrate second-order differential equations with oscillating solutions. The proposed method has high phase-lag order with small principal local truncation error coefficient. The authors given the stability analysis of the proposed method. Numerical comparisons of this new obtained method to problems with oscillating and/or periodical behavior of the solution show the efficiency of the method. 

To check the correctness of the obtained solutions we used the semi-discrete method of numerical solving of two differential equations in specific derivatives. The point of the method lies in modeling the finite-difference scheme for the space axis, the variable t being considered as continuous. In this case, the 2N ordinary differential equation system, which can be solved, for example, by the Runge-Kutta-Felberg s method with automatic choice of step along t, is obtained ... 

Since the first derivative of X(t) on the interval [T, k N exists, X(t) is continuous on such interval. If X(t) is monotoimus in the interval [7., 7(. j], the extreme values of X(t) are reached at the boundary points of the interval. Otherwise, the extreme values of X(t) can be obtained by comparing all the local extreme values found by Fermat theorem (Bronshtein et al. 1985). To calculate the value of X(t), Runge-Kutta methods can be applied for the numerical solution of the ordinary differential equations (Hairer et al. 1993, Hairer et al. 1996). 

---