Numerical methods differential equations

Example 9.1 above presents the methodology needed to obtain values for the parameters in an algebraic model that is nonlinear in the parameters. Usually, we will not be able to find an analytical solution to the differential equation model, and we will be forced to solve a differential equation model by numerical methods. This extension is intended to help you learn how to solve a model in differential equation form that is nonlinear in the parameters. That is, we will solve numerically the differential equation model (equation(9.2)) vwitten in dimensionless form where is given by equation (9.133) to obtain the parameter values by using the data in Example 9.1 above. Again, solve numerically the following differential equation model (where we have used and 2 instead of... 

The positions are obtained by numerically solving differential equations. Hence, these positions are connected in time. The positions reveal real dynamics of individual molecules. In other simulation methods, the molecular positions are not temporarily related. In other simulation methods, such as Monte Carlo simulations, the positions are generated stochastically such that a molecular configuration depends only on the previous configuration [7-10],... 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

See also Numerical Analysis and Approximate Methods and General References References for General and Specific Topics—Advanced Engineering Mathematics for additional references on topics in ordinary and partial differential equations. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

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... 

Much professional software is devoted to this problem. A diskette for sets of differential and algebraic equations with parameters to be found by this method is by Constantinides Applied Numerical Methods with Personal Computers, McGraw-Hill, 1987). 

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c 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. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of 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 Bunge Kutta method, which involves the following calculations ... 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

These differential equations are readily solved, as shown by Luyben (op. cit.), by simple Euler numerical integration, starling from an initial steady state, as determined, e.g., by the McCabe-Thiele method, followed by some prescribed disturbance such as a step change in feed composition. Typical results for the initial steady-state conditions, fixed conditions, controller and hydraulic parameters, and disturbance given in Table 13-32 are listed in Table 13-33. 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

Equations 5-110, 5-112, 5-113, and 5-114 are first order differential equations and the Runge-Kutta fourth order numerical method is used to determine the concentrations of A, B, C, and D, with time, with a time increment h = At = 0.5 min for a period of 10 minutes. The computer program BATCH57 determines the concentration profiles at an interval of 0.5 min for 10 minutes. Table 5-6 gives the results of the computer program and Figure 5-16 shows the concentration profiles of A, B, C, and D from the start of the batch reaction to the final time of 10 minutes. 

I hese equations cannot be used directly, and numerical methods are needed to compute the velocity components. The velocity components can be found by implicit differentiation and using an iterative technique.-" ... 

The population balance is a partial integro-differential equation that is normally solved by numerical methods, except for special simplified cases. Numerical solution of the population balance for the general case is not, therefore, entirely straightforward. Ramkrishna (1985) provides a comprehensive review. 

Smith, G.D., 1985. Numerical Solution of Partial Differential Equations Finite Difference Methods, 3rd edition. Clarendon Press. 

From Pm still further information about the parameters of the desorption process can be obtained. To this end, Eq. (8) must be solved. The solution, however, is accessible only in the case of desorption alone. If the contribution of the second term in Eq. (8) is appreciable, it is necessary to insert for P from Eq. (13). Thus, nonlinear differential equations result even for the most simple cases (x = 1, or the equilibrium desorption), which can be solved by numerical methods only or by iterative methods provided the second term in Eq. (8) is small. 

---