Volterra Equations

Linz, P. Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985). 

Volterra integral equations have an integral with a variable limit. The Volterra equation of the second land is... 

Equations of the first land are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. 

This integral equation is a Volterra equation of the second land. Thus the initial-value problem is eqmvalent to a Volterra integral equation of the second kind. 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

References Courant, R., and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York (1953) Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985) Porter, D., and D. S. G. Stirling, Integral Equations A Practical Treatment from Spectral Theory to Applications, Cambridge University Press (1990) Statgold, I., Greens Functions and Boundary Value Problems, 2d ed., Interscience, New York (1997). 

Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia (1985). There are methods that are analogous to the usual methods for... 

These equations are the standard form of a well-studied class of integral equations, the Volterra equation of the second kind (see, for example, Brunner and van der Houwen, 1986). Before discussing the numerical method, we draw a few simple conclusions from those equations. Using a free-electron-metal tip (that is, if in the entire energy range of interest). 

This method is known as the marching method. The accuracy of the procedure and the correctness of the program can be verified by testing it with analytically soluble Volterra equations, for example, the test problems with nonsingular convolution kernels listed on pp. 505-507 of Brunner and van der Houwen s book (1986). 

Brunner H., and van der Houwen, P. J. (1986). The Numerical Solution of Volterra Equations, North Holland, Amsterdam. 

Vibration isolation 237—250 critical damping 239 pneumatic systems 250 quality factor, Q 239 resonance excitation 241 stacked plate-elastomer system 249 transfer function 240 Virus 341 Viton 250, 270, 272 Voltage-dependent imaging 16, 17 Si(lOO) 17 Si(lll)-2X1 16 Volterra equation 310 Vortex 334 W... 

Fig. 2.3. A centre. Three solutions of the Lotka-Volterra equations (2.1.28)-(2.1.29) are presented the distinctive parameter a//3 = 1. The starting point of each trajectory is shown by...

The Lotka-Volterra equations written in the dimensionless parameters contain only several control parameters birth and death rates a, (3 and the ratio of diffusion coefficients k = Da/(Da + Z)B), 0 k < 1, i.e., Da = 2k, Db = 2(1 - k) whereas their sum is constant, DA + DB = 2. Lastly, it is also the space dimension d determining the functionals J[Z], equations (5.1.36) to (5.1.38), the Laplace operator (3.2.8) as well as the boundary condition (8.2.21) for the correlation functions of similar particles. Before discussing the results of the joint solution of the complete set of the kinetic equations, let us consider first the following statements. 

Theorem The set of memory functions K0(t),..., K (t) obey the set of coupled Volterra equations such that... 

Given an approximate A(0,the Volterra equation can be solved for C t). Our P(co) satisfies the sum rules on K co) for p0 and p2 and is therefore satisfactory to this order. It will fail to satisfy the higher-order sum rules. Nevertheless, as we pointed out, these sum rules can be built into the theory. 

Each of these postulated memories was used to solve the appropriate Volterra equation numerically for the approximate autocorrelation functions / (0 and A j(t) (see Appendix B). Three different experimental autocorrelation functions were tested the velocity autocorrelation function from both the Stockmayer and modified Stockmayer simulations and the angular momentum autocorrelation function from the modified Stockmayer simulation. The parameters needed by the postulated memory functions for each of these three autocorrelation functions are tabulated in Table IV. 

Linear viscosity is that when the function is splitted in both creep response and load. All linear viscoelastic models can be represented by a classical Volterra equation connecting stress and strain [1-9] ... 

INTEGRAL EQUATIONS, F.G. Tricomi. Authoritative, well-written treatment of extremely useful mathematical tool with wide applications. Volterra Equations, Fredholm Equations, much more. Advanced undergraduate to graduate level. Exercises. Bibliography. 238pp. 5k x 8k. 64828-1 Pa. 6.95... 

If adsorption is fast but not sufficiently strong to justify the assumption C0 0, then C0 will, at any instant, be determined by the adsorption isotherm (2.103). This boundary condition leads to mathematical problems the integral equation resulting from (2.110) then becomes a Volterra equation. This has been solved for only some very simple isotherms. Delahay and Trachtenberg [203] solved it for the Henry isotherm (2.104), the solution being... 

When we speak of mathematical models for biology, we usually refer to formulae (such as the Hardy-Weinberg theorem, or the Lotka-Volterra equations) that effectively describe some features of living systems. In our case, embryonic development is not described by integrals and deconvolutions, and the formulae of the reconstruction algorithms cannot be a direct description of what happens in embryos. There is however another type of mathematical model. The formulae of energy, entropy and information, for example, apply to all natural processes, irrespective of their mechanisms, and at this more general level there could indeed be a link between reconstruction methods and embryonic development. For our purposes, in fact, what really matters are not the formulae per se, but... 

The physical separation between genotype and phenotype has an extraordinary consequence, because mental genotypes can be directly instructed by mental phenotypes, and this means that cultural heredity is based on a transmission of acquired characters. Cultural inheritance, in other words, is transmitted with a Lamarckian mechanism, whereas biological inheritance relies on a Mendelian mechanism which is enormously slower. As a result, cultural evolution is much faster than biological evolution, and almost all differences between biology and culture can be traced back to the divide that exists in their hereditary mechanisms. The discovery that human artifacts (i.e. cultural phenotypes) obey the Lotka-Volterra equations has two outstanding consequences. The first is that selection accounts for all types of adaptive evolution natural selection is the mechanism hy which all phenotypes - biological as well as cultural - diffuse in the world. 

Example 13.8 Prey-predator system Lotka-Volterra model The Lotka-Volterra predator and prey model provides one of the earliest analyses of population dynamics. In the model s original form, neither equilibrium point is stable the populations of predator and prey seem to cycle endlessly without settling down quickly. The Lotka-Volterra equations are... 

To investigate the stability properties of the Lotka-Volterra equation in the vicinity of the equilibrium point (X, Y) = (0, 0), we linearize the equations ofXand Yappearing on the right side of Eqs. (13.63) and (13.64). These functions are already in the form of Taylor series in the vicinity of the origin. Therefore, the linearization requires only that we neglect the quadratic terms in XT, and the Lotka-Volterra equations become... 

---