8/14/2019 18.2 Volterra Equations

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786 Chapter 18. Integral Equations and Inverse Theory

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18.2 Volterra Equations

Let us now turn to Volterra equations, of which our prototype is the Volterraequation of the second kind,

f (t) = t

aK (t, s )f (s) ds + g(t) (18.2.1 )

Mostalgorithms forVolterra equationsmarch out from t = a , building up thesolutionas they go. In this sense they resemble not only forward substitution (as discussedin 18.0), but also initial-value problems for ordinary differential equations. In fact,many algorithms for ODEs have counterparts for Volterra equations.

The simplest way to proceed is to solve the equation on a mesh with uniformspacing:

t i = a + ih, i = 0 , 1, . . . , N, h b a

N (18.2.2 )

To do so, we must choose a quadrature rule. For a uniform mesh, the simplestscheme is the trapezoidal rule, equation (4.1.11):

t i

aK (t i , s )f (s) ds = h 12 K i 0 f 0 +

i 1

j =1

K ij f j + 12 K ii f i (18.2.3 )

Thus the trapezoidal method for equation (18.2.1) is:

f 0 = g0

(1 12 hK ii )f i = h12 K i 0 f 0 +

i 1

j =1

K ij f j + gi , i = 1 , . . . , N (18.2.4 )

(For a Volterra equation of the rst kind, the leading 1 on the left would be absent,and g would have opposite sign, with corresponding straightforward changes in therest of the discussion.)

Equation (18.2.4) is an explicit prescription that gives the solution in O(N 2 )operations. Unlike Fredholm equations, it is not necessary to solve a system of linearequations. Volterra equations thus usually involve less work than the correspondingFredholm equations which, as we have seen, do involve the inversion of, sometimeslarge, linear systems.

The efciency of solving Volterra equations is somewhat counterbalanced bythe fact that systems of these equations occur more frequently in practice. If weinterpret equation (18.2.1) as a vector equation for the vector of m functions f (t),

then the kernel K (t, s ) is an m m matrix. Equation (18.2.4) must now also beunderstood as a vector equation. For each i , we have to solve the m m set of linear algebraic equations by Gaussian elimination.

The routine voltra below implements this algorithm. You must supply anexternal function that returns the kth function of the vector g(t) at the point t , andanother that returns the (k, l ) element of the matrix K (t, s ) at (t, s ). The routinevoltra then returns the vector f (t) at the regularly spaced points t i .

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