Functional Operators, Volume 1: Measures and Integrals.

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The most flexible way of spatial integration is to add an additional PDE interface.

Measure Theory contents

The task can be formulated in terms of the PDE. The easiest interface to implement this equation is the Coefficient Form PDE interface, which only needs the following few settings:. How to use an additional physics interface for spatial integration. The dependent variable u represents the antiderivative with respect to x and is available during calculation and postprocessing.

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Besides flexibility, a further advantage of this method is accuracy, because the integral is not obtained as a derived value, but is part of the calculation and internal error estimation. We have already mentioned the Data Series Operations, which can be used for time integration. Another very useful method for time integration is provided by the built-in operators timeint and timeavg for time integration or time average, respectively. They are readily available in postprocessing and are used to integrate any time-dependent expression over a specified time interval. In our example we may be interested in the temperature average between 90 seconds and seconds, i.

The following surface plot shows the resulting integral, which is a spatial function in x,y :. How to use the built-in time integration operator timeavg. Similar operators are available for integration on spherical objects, namely ballint , circint , diskint , and sphint. If temporal integrals have to be available in the model, you need to define them as additional dependent variables. Suppose, for example, that at each time step, the model requests the time integral from start until now over the total heat flux magnitude, which measures the accumulated energy.

The source term of this domain ODE is the integrand , as shown in the following figure. How to use an additional physics interface for temporal integration.


What is the benefit of such a calculation? The integral can be reused in another physics interface, which may be influenced by the accumulated energy in the system. Moreover, it is now available for all kinds of postprocessing, which is more convenient and faster than built-in operators. For an example, check out the Carbon Deposition in Hetereogeneous Catalysis model , where a domain ODE is used to calculate the porosity of a catalyst as a time-dependent field variable in the presence of chemical reactions. So far, we have shown how to integrate solution variables during calculation or in postprocessing.

We have not yet covered integrals of analytic functions or expressions.

Functional Operators: Measures and integrals, Volume 1

To this end, COMSOL provides the built-in operator integrate expression , integration variable , lower bound , upper bound. The expression might be any 1D function, such as sin x. The second argument specifies over which variable the integral is calculated. Note that the operator can also handle analytic functions, which need to be defined in the Definitions node of the current component.

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Publication date Topics jordan measure , measurable functions , lebesgue integral , L2 space , fubini's theorem , mathematics , measure theory , hilbert space , riesz-fisher theorem , orthogonal functions , integral equations , linear functions , self-adjoint operators , fourier series , semirings Collection opensource Language English.

Measure of Plane Sets. Systems of Sets. Continuations of Jordan Measures.

Theoretical Techniques

Countable Additivity. General Problem of Continuation of Measures. Lebesgue Continuation of Measures in the General Case. Definition and Basic Properties of Measurable Functions. Sequences of Measurable Functions. Different Types of Convergence.