Integration factor method

To find the dependence of Cb on time we use the integrating factor method. Write Eq. (3-22) as... 

The differential equations (36) and (37) are solved by the integrating factor method, with the boundary condition C = C0 at x = 0. 

Use the integrating factor method to find the general solutions to first-order differential equations linear in y... 

Finding general solutions to linear first order differential equations using the integrating factor method. 

This equation is in the proper form for solution by the integrating factor method, that is ... 

This first-order initial-value problem can easily be solved by the use of the integration factor method. That is. 

We solve (13.73) subject to (13.77) using the integrating factor method. The integrating factor will be given by the exponential of the integral of the term multiplying n(v,t) on the left-hand side of (13.73) ... 

This equation can be solved using the Integrating factor method, with the Initial condition that at t - 0, C = Cq, the preload concentration of toluene In the process tank. The resulting expression for C Is QfCfCV , + DVh) QfCf(Vw + DVh) 1... 

As it stands, the above equation is classified as a linear, inhomogeneous equation of first order, which in general must be solved using the so-called Integrating-Factor method, as we discuss later in Chapter 2 (Section 2.3). 

Where - = +1 V is substantial derivative length wise of microscopic velocity. Equation (18) that mentioned is a lineal ODE of first order, which can be integrated over a time step of 8t. Using the integral factor method is obtaining... 

Where k s is the ratio of the constant in Eq. (20) to that in Eq. (19). This equation is solved with the integrating factor method C) and the above-mentioned assumption that k < k2 to give... 

Zhao, S., Ovadia, J., Liu, X., Zhang, Y.-T., Nie, Q. Operator splitting implicit integration factor methods for stiff reaction-diffusion-advection systems. J. Comput. Phys. 230, 5996-6009 (2011)... 

This differential equation can be solved via the integrating factor method. The solution is Rearranging and taking the log of both sides... 

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have... 

The production rates obtained after differentiation are then regressed by a postulated rate equation on parameters such as preexponential factors and activation energies. This method is termed the differential method since the fit of the rate expression is tested to the data directly, without any integration. This method is useful in relatively complicated situations. However, the required differentiation of Xa versus W/ Fao data can be a source of errors. To avoid this, the same set of data can be analyzed by the so-called integral method, i.e., by selecting a rate equation and integrating it (Eqn. (8.26)). The integral method can only test one particular rate expression at a time, while the differential method can be used to develop a rate equation to fit the data [16]. 

There is no general method for finding an integrating factor, although we will discuss a method that will work for a particular class of differential forms in Chapter 8. However, it is true that if a differential possesses one integrating factor, there are infinitely many integrating factors for that differential. 

The dimensionless shape factor for the isothermal rectangular annulus is derived from the correlation equation of Schneider [89], who obtained accurate numerical values of the thermal constriction resistance of doubly connected rectangular contact areas by means of the boundary integral equation method ... 

The data represented here should be viewed as a conservative estimate of the sensitivity of the INTEGRAL Imager. Methods of preferential event selection, with the aim of increasing the polarimetric modulation factor are currently under investigation. It is believed that with these analysis methods the sensitivity can be improved by at least a factor of two. 

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