The equations in the unknown f(t) of the form
$$\int\limits_0^t {k\left( {t,\tau } \right)f\left( \tau \right)d\tau \, = \,g\left( t \right)} \,\,\,\,{\rm{and}}\,\,\,\,f\left( t \right)\, = \,g\left( t \right) + \int\limits_0^t {k\left( {t,\tau } \right)f\left( \tau \right)d\tau }$$
are known as Volterra linear integral equations of the first and second kind respectively. These integral equations are of the convolution type, provided the kernel k(t, τ) is a function of (t − τ) only. These latter equations can be changed into algebraic equations by the L-transformation, invoking the Convolution Theorem. Applying the inverse Laplace transformation to the solution of the algebraic equation, one obtains the solution of the integral equation.


Integral Equation Series Representation Finite Interval Original Space Convolution Theorem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1974

Authors and Affiliations

  • Gustav Doetsch
    • 1
  1. 1.Emeritus of MathematicsUniversity of FreiburgFreiburg i. Br.Germany

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