Symbolic toolbox

Solutions for the integration of ODE s, such as the ones given in equations (3.76), are not always readily available. For non-specialists it is difficult to determine if there is an explicit solution at all. The symbolic toolbox (which is not contained in the standard Matlab) provides very convenient means to integrate systems of differential equations and also to test whether there is an explicit solution. As an example the reaction 2A —-—> B ... 

Note that Matlab s symbolic toolbox demands lower case characters for species names. 

Below, the attempt to use the symbolic toolbox for the integration of a slightly more complex mechanism ... 

While there is an analytical solution for this mechanism, the formula for the calculation of the concentration profiles for A and B is fairly complex, involving the tan and atan functions (according to Matlab s symbolic toolbox). We use it to demonstrate the basic ideas of numerical integration. 

In the above example, equation (5.34), it is relatively straightforward to determine the eigenvalues of K. In the example (5.35) it is much more difficult to develop the equations. The Symbolic Toolbox of Matlab can be employed for the task. 

The Symbolic Toolbox can even cope with initial concentrations and thus delivers the equations for the concentration profiles. 

Solutions for the integration of ODEs such as those given in Equation 7.5 are not always readily available. For nonspecialists, it is difficult to determine whether there is an explicit solution at all. MATLAB s symbolic toolbox provides a very convenient means of producing the results and also of testing for explicit solutions of ordinary differential equations, e.g., for the reaction 2A — B, as seen in MATLAB Example 7.2. (Note that MATLAB s symbolic toolbox demands lowercase characters for species names.)... 

In this section we analyze processes involving the second-order A + B —> 2P reaction. Such processes have been studied, among others, by Luyben andTyreus [10]. It has been noticed [11] that certain control structures lead to state multiplicity and instability. Here, we use dimensionless models to derive general feasibility and stability conditions. The reader is encouraged to check carefully the balance equations, writing them first in the dimensional form, and then deriving the dimensionless versions. To solve these equations, software such as Maple or the symbolic toolbox of Matlab can be used. 

Can the equation be solved analytically If yes, perform the calculations. If you have trouble integrating, use the int function in MATLAB which is part of the symbolic toolbox. 

We wish to determine the local stability of the CSTR system of section 4.3. To do this, we first linearize the nonlinear problem about the steady-state condition. You might want to use the symbolic toolbox of MATLAB to help in this linearization. This converts the problem to a linear set of differential equations... 

Analytical solutions of some ODEs can be obtained with dsolve. As an example, a solution to a particle settling at low particle Reynolds number can be obtained with the Symbolic toolbox in MATLAB ... 

In this expression, the n are the stoichiometric coefficients in the chemical equation and the symbol 2 (sigma) means a sum. The first sum is the total enthalpy of formation of the products. The second sum is the similar total for the reactants. Toolbox 6.2 explains in more detail how to use this expression. 

Lewis Structures Lewis structures are one of the most useful and versatile tools in the chemist s toolbox. G. N. Lewis reported this model for chemical bonding in 1902. Lewis structures are nonmathematical models that allow us to qualitatively describe the chemical bonding in a molecule and then gain insights about the physical and chemical properties we can expect of that molecule. Don t discount the power of Lewis structures just because the underlying mathematics isn t evident. In a Lewis structure, the atoms are represented by their chemical symbol. Lines between atoms represents shared pairs of electrons in covalent bonds. Valence electrons that are not used for covalent bonds are lone pairs, and they are represented as pairs of dots on the atom. 

The proposed matrix-based approach is illustrated by manual derivation of results for small, well-known examples. For more complex system models, software such as CAMP-G/MATLAB together with the Symbolic Math Toolbox can be used. 

Performing these operations by hand is practically hardly feasible even for small systems. However, software programs such as CAMP-G, MATLAB , and the Symbolic Math Toolbox can set up the matrices of the state space models, perform multiplications of matrix entries, and build the sum of terms. 

Existing software such as CAMP-G/MATLAB supported by the Symbolic Math Toolbox can derive equations from the bond graph and from its associated incremental bond graph and can build the matrices of the state space equations and the output equations for both bond graphs in symbolic form. 

In the case of linear system models, the combination of CAMP-G, MATLAB, and the Symbolic Math Toolbox can generate state space matrices as well as transfer functions in symbolic form from a bond graph. MATLAB in conjunction with the Symbolic Math Toolbox can also be used for the incremental bond graph approach presented in Chapter 4. 

The computer-generated transfer function for the voltage across the capacitor crosses two different energy domains without separation since the model is all together. The transfer function is obtained in one step in symbolic form. CAMPG generated the code for the A, B, C, D matrices which are displayed in MATLAB. Any other transfer function for the efforts and flow output variables can be obtained. More details are presented in [11]. At this point, the computer-generated model becomes so versatile that all the linear control theory operations implemented in the MATLAB Control Systems Toolbox can be used on the entire mechatronics model. 

Mathematica has packages available for calling MATLAB from within Mathematica and vice versa. MATLAB also has toolboxes that incorporate some of the symbolic capabilities of MAPLE, as pointed out in the hterature [10]. 

Mathematical modeling. SIMULINK for simulating nonlinear dynamic systems. Toolboxes for symbolic math, neural networks, statistics, spline analysis, signal processing, and image processing. PCs (Windows) and Macintosh. 

Using MATLAB s Symbolic Math Toolbox, you can carry out algebraic or symbolic calculations such as factoring polynomials or solving algebraic equations. Type help symbolic to make sure that the Symbolic Math Toolbox is installed on your system. 

The MATLAB software consists of a basic package of mathematical routines as well as optional toolboxes that cover specific engineering application areas such as process control, optimisation, signal processing and symbolic mathematics. It also has a toolbox that makes available many of the Numerical Algorithms Group (NAG) routines. These routines are some of the best implementations of numerical methods that are currently available. MATLAB offers a convenient computing environment for the solution of process simulation problems, and is used here. 

We use MATLAB R201 lb and the Symbolic Math Toolbox V5.7 for the calculation of the average delays as described in Sect. 4.1. Figures 4.1a and b show Fw, for a node moving in a left direction with r = 0 and t = oo respectively. The main difference... 

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