Integrodifferential equation

Here we treat it as a model one. However, the arguments about this matter can result in the design of interesting experiments, whose aims and scope are to test and improve admisssible schemes for rather complicated equations of acoustics, kinematic integrodifferential equations of neutron transfer, nonlinear equations of gas dynamics, etc. Because of the enormous range and variety of problems dealt with by mathematical physics, the contents of this section would be of the methodological merit. 

Kapur (K4) has shown that this integrodifferential equation admits a similarity solution given in Eqs. (49)-(51), namely,... 

Combining Eqs. (76) and (77) results in a total integrodifferential equation in a single similarity variable p... 

The integrated expression relating radius to time and temperature is a nonlinear, integrodifferential equation. It can be put into tractable forms by imagining four overlapping growth periods. 

We find from here that u(x, t) is a solution of the integrodifferential equation... 

Vol. 3. Ill-Posed Problems for Integrodifferential Equations in Mechanics and Electromagnetic Theory Frederick Bloom... 

The above equation is an integrodifferential equation that has an unusual structure. Here (co2)1 2 is the frequency of the free oscillator and y is the damping constant. The fourth term on the left-hand side of Eq. (226) has the form of the memory kernel, and its strength is controlled by the dimensionless coupling constant X which contains the contribution from the vertex function. 

This was the idea behind concept (14b) in Ref.4). The corresponding formulation of Eq. (15i) of Ref.4), however, was unhappily chosen Eq. (15f)ofthis paper should have been used. If the kinetics of separation were explicitly introduced into the transport Equation of PDC (instead of the implicit concept (I7a-c) of the flow-equilibrium), an integrodifferential equation more complicated than (41 a-b) would be obtained, which could hardly be solved analytically... 

An analytical integration of an integrodifferential equation under a singular time boundary is always a complicated matter. The treatment of the method, based on a representation of the delta functional as a Fourier transform, and working in the complex plane, would be out of place in this report. It can be found in detail in Ref. 7) where also the solution obtained is discussed. It is shown that this solution is especially simple if the elution curves show a positive skewness, i.e. if they are tailed on the right-hand-side of their maximum (this is always true in PDC and GPC). A renormalization of the found concentration profile and a recalculation of the coordinates (z, t) to the elution volumina (V, V) then yield the spreading function of the considered column (Greschner 7))... 

The elution and migration effects are also found in PDC, but not the compression effect, since no precipitant and temperature gradients are applied. Unlike BWF, no simple transport Equation is applied in PDC because the replacement of the linear Eq. (3b) by the non-linear (17c) would lead to an integrodifferential equation similar to (41 a-b), but more complicated, if some explicit formulation were used instead of the implicit one, based on a flow-equilibrium and on a perturbation calculus, applied to an integrated transport Equation... 

The prediction of the MWD of emulsion polymers proved to be a relatively intractable problem even after the advent of the Harkins-Smith-Ewart theory. Perhaps the most successful early attack on the problem was that of Katz, Shinnar and Saidel (2). They considered only two microscopic events entry and bimolecular termination by combination. Their theory resulted in a set of partial integrodifferential equations, whose numerical solution provided the lower moments of the molecular weight distribution function. Other attempts to predict the MWD of emulsion polymers include those of Parts and Wat ter son (3 ), Sundberg and Eliassen (4), Min and Ray (5) and Gardon (6). 

These choices give the simple integrodifferential equation... 

The fast desorption of CO in CO/Cu(OOf) has been measured [33] and also calculated. [30,31] The collision induced vibrational excitation and following relaxation of CO on Cu(001) has also been experimentally explored using time-of-flight techniques, and has been analyzed in experiments [34] and theory. [23,32] Our previous treatment of instantaneous electronic de-excitation of CO/Cu(001) after photoexcitation is extended here to include delayed vibrational relaxation of CO/Cu(001) in its ground electronic state. We show results for the density matrix, from calculations with the described numerical procedure for the integrodifferential equations. 

Density operator equations were converted into coupled integrodifferential equations suitable for numerical processing, and an extended Runge-Kutta algorithm has been implemented for solving the matrix equations in diadic form. A similar procedure can be followed for the original density matrix. 

As an illustration of the numerical treatment, the instantaneous dissipation due to fast electronic motions was constructed from the Lindblad expression in the treatment of CO/Cu(001), from decay and transition rates. The delayed dissipation, present for slow atomic vibrations of the medium, was given in terms of a memory kernel in the integrodifferential equation, calculated to second order in the coupling of the adsorbate and its environment. 

It should be noted that the integrodifferential equations equivalent to Eqs. (3.168) were first obtained with the fully renormalized kinetic theory of Yang, Lee, and Shin (YLS) [132]. However, they had not confined themselves to the original integral equations (3.168) but reduced them to the differential rate equations by the incorrect substitution used instead of (3.164) ... 

From the set of equations (3.234), we obtain the single integrodifferential equation for exciplex concentration ... 

When the B particles are present in great excess, the following integrodifferential equations linear in c describe the reversible contact reaction [47] ... 

This orthogonality condition is implemented in close-coupling theory by introducing Lagrange multipliers x s into the radial integrodifferential equations,... 

Equations (14.8) and (14.11) are two coupled integrodifferential equations which must be solved simultaneously for unknowns, 0( i) and P( i). Analytical solutions are unlikely, but we can solve numerically for prescribed values of e, y and Nc. 

Series RLC Circuit. Consider the series RLC circuit shown in Fig. 9.2. The three impedances add, so the overall impedance is analyzed by setting up an integrodifferential equation... 

With the help of Eq. (4) (after an integration by parts of the last term of its right-hand side), Eq. (5) can be reformulated as a closed integrodifferential equation for x(f) ... 

To circumvent the problem of many equations, it is possible to represent the coagulation equation as a nonlinear integrodifferential equation of the form... 

For the applications discussed in the following sections, the preceding conservation equations are supplemented by ttje ideal gas equation of state, in which Pg (not the fluid density Pf) enters. Since fj appears in each of the conservation equations, it is apparent that they are coupled to the spray equation, which therefore must also be included to obtain a complete set of integrodifferential equations describing spray combustion. 

Since the initial drop-size distribution is known, the integral in equation (92) is a knownS unction of (> /w) dx, [see equation (91)]. Equation (92) is therefore an integrodifferential equation for determining the rate of disappearance of the liquid. 

Boltzmann s name is best reserved for the form that this equation takes when only binary collisions contribute to dfi/Sr, an explicit, quadratically nonlinear, integrodifferential equation. 

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