Integral equations reduced

Exercise. Show that under certain conditions this integral equation reduces to a wave equation with renormalized wave number... 

The energy integral equation reduces for low-speed flow over a flat plate to... 

Consider the integral equation li[/(r)] = 0 for a function /(r) defined in the 3D space, discretized on a grid of k points. The solution is thus represented by a vector f r-k), and the integral equation reduces to a set of k nonlinear equations zeroing the values of the residual li[/(r)] at the points rjt, that is zeroing the residual vector... 

The integral equation approach has also been explored in detail for electrolyte solutions, with the PY equation proving less usefiil than the HNC equation. This is partly because the latter model reduces cleanly to the MSA model for small h 2) since... 

The first and second integrals on the right-hand side of this equation evaluate to 1 because the atomic wave functions are assumed to be normalized. Therefore, the right-hand side of the equation reduces to... 

This integral equation for Q reduces in the limit of instantaneous collisions (t (Aco)-1) to a closed differential equation for 1ITZ ... 

Equation (24) can be derived from the theory of hyperspherical harmonics and Gegenbauer polynomials but for readers unfamiliar with this theory, the expansion can be made plausible by substitution into the right-hand side of equation (23). With the help of the momentum-space orthonormality relations, (17), it can then be seen that right-hand side of (23) reduces to the left-hand side, which must be the case if the integral equation is to be satisfied. Let us now consider an electron moving in the attractive Coulomb potential of a collection of nuclei ... 

Koga and others [10] have shown that the loss of accuracy resulting from tmncation of the basis set can be reduced by replacing the one-electron secular equation, (34), with an equation based on a second iteration of the integral equation (31). The second-iterated integral equation has the form ... 

If the density is constant (e = 0), then the integrated forms of the design equation reduce to... 

Using integral equation (4) the integral then reduces to ... 

Clearly, efficiency of the symmetry reduction procedure is subject to our ability to integrate the reduced systems of ordinary differential equations. Since the reduced equations are nonlinear, it is not at all clear that it will be possible to construct their particular or general solutions. That it why we devote the first part of this subsection to describing our technique for integrating the reduced systems of nonlinear ordinary differential equations (further details can be found in Ref. 33). 

This is an equation for F(z, t 1,0) alone, which determines the probability generating function and hence the distribution, once y(i) is known. The treatment of the branching process is thereby reduced to solving a nonlinear integral equation. Unfortunately this can only be done explicitly for very few choices of y(r). 

It was the simplified integrated Equations 7 and 8 that were found sufficient for handling the experimental data. However, the (act that B is independent o( S explains the qualitative observation that for slower sorptions —e.g., on acetylated coals—deviations (rom the simplified equations occurred at lower (ractions o( reaction than for (aster reactions. This observation is consistent with the conclusion that acetylation merely reduces the concentration o( all surface sites, and the acetate group itseK is rather inert to methanol sorption. 

Due to a similarity of reaction stages in the Lotka and Lotka-Volterra models the equations for the pop and pop remain the same as in Section 8.2. Other kinetic equations are slightly simplified, a number and multiplicity of integrals are reduced. 

The integrated continuity equation is a weaker form of the full continuity equation. This is noticed in numerical solutions of mold filling problems, where continuity is never fully satisfied. However, this violation of continuity is insignificant and will not hinder the solution of practical mold filling problems. The integrated continuity equation reduces to... 

The fiber is suspended in the liquid, which means that due to small time scales given by the pure viscous nature of the flow, the hydrodynamic force and torque on the particle are approximately zero [26,51]. Numerically, this means that the velocity and traction fields on the particle are unknown, which differs from the previous examples where the velocity field was fixed and the integral equations were reduced to a system of linear equations in which velocities or tractions were unknown, depending on the boundary conditions of the problem. Although computationally expensive, direct integral formulations are an effective way to find the velocity and traction fields for suspended particles using a simple iterative procedure. Here, the initial tractions are assumed and then corrected, until the hydrodynamic force and torque are zero. 

For Newtonian fluids with s = 1, this equation reduces to the classic parabolic profile, (b) The volumetric flow rate is obtained by integrating Eq. E3.4-12. 

This approach, based on a complex-valued realization of the PCM algorithm, reduces to a pair of coupled integral equations for real and imaginary parts of apparent charge density for tr(f,to) [13]. An alternative technique avoiding explicit treatment of the complex permittivity has been also derived [14,15]. The kernel K(f,f, t) of operator K does not appear explicitly. However, its matrix elements can be computed for any pair of basis charge densities p1(r) and p2(r) px k p = Jp2(j) (r, f)d3r, where tp(r, t), given by Equation (1.137), corresponds to p(r) = p2(r). 

However, MET is not a unique theory accounting for multiparticle effects. There are some others competing between themselves, but they all can be reduced to the integral equations of IET distinctive only by their kernels. Depending in a different way on the concentration of quenchers c, the kernels of all contact theories of irreversible quenching coincide with that of IET in the low concentration limit (c —> 0) [46], IET of the reversible dissociation of exciplexes is also the common limit for all multiparticle theories of this reaction, approached at c = 0 [47], This universality and relative simplicity of IET makes it an irreplaceable tool for kinetic analysis in dilute solutions. 

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