Volterra Integral Equations

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

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. 

The relationship between creep and relaxation experiments is more complex. The complexity of the transforms tends to increase when stress and strain lead experiments are transformed in the time domain. This can be tackled in a number of ways. One mathematical form relating the two is known as the Volterra integral equation which is notoriously difficult to evaluate. Another, and perhaps the conceptually simplest form of the mathematical transform, treats the problem as a functional. Put simply, a functional is a rule which gives a set of functions when another set has been specified. The details are not important for this discussion, it is the result which is most useful ... 

A further integration, followed by a change in the order of integration, converts this into a Volterra integral equation, which can be solved formally using the Picard method. However, this does not appear to be very practical or efiicient, except in those rare cases where V is small and the Picard series converges rapidly. 

The solution of the problem can be obtained via the following system of non-linear Volterra Integral Equations [94] ... 

The monograph by Linz (1985) provides a good theoretical and practical treatment of Volterra integral equations. In what follows, we list the solutions in accordance with the kernel type. 

Numerical Solution of Volterra Integral Equations of the Second Kind... 

It is the first time in the present paper that the uptake curves from a piezometric apparatus have been simulated by the solution of the Volterra integral equation [9,10] which reflects in detail the interaction of the sorption kinetics with the apparatus. This approach enabled us to get kinetic data with a high accuracy. 

This adsorption model can be solved analytically using Laplace Transforms (Hansen 1961, Miller 1983) but the result is a non-linear Volterra integral equation similar to the Ward Tordai equation (4.1) ... 

It can be concluded that the dynamic adsorption equation (8.85) in the limiting cases leads to the same results as obtained by other methods. Expressed in terms of the variables x = cos0 y(x) = sin 0 c(a, 0), Eq. (8.85) is transformed into the Volterra integral equation. 

In some cases, for numerical calculation of nonlinear equations, one can use a fact that fractional derivative is based on a convolution integral, the number of weights used in the numerical approximation to evaluate fractional derivatives. In addition, one can apply predictor-corrector formula for the solution of systems of nonlinear equations of lower order. This approach is based on rewriting the initial value problem (15.68) and (15.69) as an equivalent fractional integral equation (Volterra integral equation of the second kind)... 

Micke, A., and Bulow, M., Application of Volterra integral equations to the modelling of the sorption kinetics of multicomponent mixtures in porous media Fundamentals and elimination of apparatus effects. Gas Sep. Purif.,4(3), 158-170(1990). 

The Volterra integral equation (7.3.28) can be solved by successive substitution (or otherwise) for the unknown function / (O, t) to obtain... 

Bieniasz LK (2010) Automatic simulation of cyclic voltammograms by the adaptive Huber method for weakly singular second kind Volterra integral equations. Electrochim Acta 55 721-728... 

Bieniasz LK (2011) Automatic simulation of electrochemical transients at cylindrical wire electrodes, by the adaptive Huber method for Volterra integral equations. J Electroanal Chem 662 371-378... 

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