Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Volterra integral equation
open-in-new
<h2 id="conversion-of-volterra-equation-of-the-first-kind-to-the-second-kind">Conversion of Volterra equation of the first kind to the second kind</h2> <p>A linear Volterra equation of the first kind can always be reduced to a linear Volterra equation of the second kind, assuming that K ( t , t ) ≠ 0 {\displaystyle K(t,t)\neq 0} . Taking the derivative of the first kind Volterra equation gives us: d f d t = ∫ a t ∂ K ∂ t x ( s ) d s + K ( t , t ) x ( t ) {\displaystyle {df \over {dt}}=\int _{a}^{t}{\partial K \over {\partial t}}x(s)ds+K(t,t)x(t)} Dividing through by K ( t , t ) {\displaystyle K(t,t)} yields: x ( t ) = 1 K ( t , t ) d f d t − ∫ a t 1 K ( t , t ) ∂ K ∂ t x ( s ) d s {\displaystyle x(t)={1 \over {K(t,t)}}{df \over {dt}}-\int _{a}^{t}{1 \over {K(t,t)}}{\partial K \over {\partial t}}x(s)ds} Defining f ~ ( t ) = 1 K ( t , t ) d f d t {\textstyle {\widetilde {f}}(t)={1 \over {K(t,t)}}{df \over {dt}}} and K ~ ( t , s ) = − 1 K ( t , t ) ∂ K ∂ t {\textstyle {\widetilde {K}}(t,s)=-{1 \over {K(t,t)}}{\partial K \over {\partial t}}} completes the transformation of the first kind equation into a linear Volterra equation of the second kind. </p> <h2 id="numerical-solution-using-trapezoidal-rule">Numerical solution using trapezoidal rule</h2> <p>A standard method for computing the numerical solution of a linear Volterra equation of the second kind is the <a href="/facts/Trapezoidal_rule/KEg8cLGa">trapezoidal rule</a>, which for equally-spaced subintervals Δ x {\displaystyle \Delta x} is given by: ∫ a b f ( x ) d x ≈ Δ x 2 [ f ( x 0 ) + 2 ∑ i = 1 n − 1 f ( x i ) + f ( x n ) ] {\displaystyle \int _{a}^{b}f(x)dx\approx {\Delta x \over {2}}\left[f(x_{0})+2\sum _{i=1}^{n-1}f(x_{i})+f(x_{n})\right]} Assuming equal spacing for the subintervals, the integral component of the Volterra equation may be approximated by: ∫ a t K ( t , s ) x ( s ) d s ≈ Δ s 2 [ K ( t , s 0 ) x ( s 0 ) + 2 K ( t , s 1 ) x ( s 1 ) + ⋯ + 2 K ( t , s n − 1 ) x ( s n − 1 ) + K ( t , s n ) x ( s n ) ] {\displaystyle \int _{a}^{t}K(t,s)x(s)ds\approx {\Delta s \over {2}}\left[K(t,s_{0})x(s_{0})+2K(t,s_{1})x(s_{1})+\cdots +2K(t,s_{n-1})x(s_{n-1})+K(t,s_{n})x(s_{n})\right]} Defining x i = x ( s i ) {\displaystyle x_{i}=x(s_{i})} , f i = f ( t i ) {\displaystyle f_{i}=f(t_{i})} , and K i j = K ( t i , s j ) {\displaystyle K_{ij}=K(t_{i},s_{j})} , we have the system of linear equations: x 0 = f 0 x 1 = f 1 + Δ s 2 ( K 10 x 0 + K 11 x 1 ) x 2 = f 2 + Δ s 2 ( K 20 x 0 + 2 K 21 x 1 + K 22 x 2 ) ⋮ x n = f n + Δ s 2 ( K n 0 x 0 + 2 K n 1 x 1 + ⋯ + 2 K n , n − 1 x n − 1 + K n n x n ) {\displaystyle {\begin{aligned}x_{0}&=f_{0}\\x_{1}&=f_{1}+{\Delta s \over {2}}\left(K_{10}x_{0}+K_{11}x_{1}\right)\\x_{2}&=f_{2}+{\Delta s \over {2}}\left(K_{20}x_{0}+2K_{21}x_{1}+K_{22}x_{2}\right)\\&\vdots \\x_{n}&=f_{n}+{\Delta s \over {2}}\left(K_{n0}x_{0}+2K_{n1}x_{1}+\cdots +2K_{n,n-1}x_{n-1}+K_{nn}x_{n}\right)\end{aligned}}} This is equivalent to the <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> equation: x = f + M x ⟹ x = ( I − M ) − 1 f {\displaystyle x=f+Mx\implies x=(I-M)^{-1}f} For well-behaved kernels, the trapezoidal rule tends to work well. </p> <h2 id="application-ruin-theory">Application: Ruin theory</h2> <p>One area where Volterra integral equations appear is in <a href="/facts/Ruin_theory/Jq3MaHIS">ruin theory</a>, the study of the risk of insolvency in actuarial science. The objective is to quantify the probability of ruin ψ ( u ) = P [ τ ( u ) < ∞ ] {\displaystyle \psi (u)=\mathbb {P} [\tau (u)<\infty ]} , where u {\displaystyle u} is the initial surplus and τ ( u ) {\displaystyle \tau (u)} is the time of ruin. In the <a href="/facts/Ruin_theory/Jq3MaHIS">classical model</a> of ruin theory, the net cash position X t {\displaystyle X_{t}} is a function of the initial surplus, premium income earned at rate c {\displaystyle c} , and outgoing claims ξ {\displaystyle \xi } : X t = u + c t − ∑ i = 1 N t ξ i , t ≥ 0 {\displaystyle X_{t}=u+ct-\sum _{i=1}^{N_{t}}\xi _{i},\quad t\geq 0} where N t {\displaystyle N_{t}} is a <a href="/facts/Poisson_point_process/8uphjAUc">Poisson process</a> for the number of claims with intensity λ {\displaystyle \lambda } . Under these circumstances, the ruin probability may be represented by a Volterra integral equation of the form<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: ψ ( u ) = λ c ∫ u ∞ S ( x ) d x + λ c ∫ 0 u ψ ( u − x ) S ( x ) d x {\displaystyle \psi (u)={\lambda \over {c}}\int _{u}^{\infty }S(x)dx+{\lambda \over {c}}\int _{0}^{u}\psi (u-x)S(x)dx} where S ( ⋅ ) {\displaystyle S(\cdot )} is the <a href="/facts/Survival_function/jkdCmDKq">survival function</a> of the claims distribution. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Fredholm_integral_equation/euHGJbQo">Fredholm integral equation</a></li> <li><a href="/facts/Integral_equation/LCu4Imfq">Integral equation</a></li> <li><a href="/facts/Integro-differential_equation/l0YHwgUY">Integro-differential equation</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Traian Lalescu, <i>Introduction à la théorie des équations intégrales. Avec une préface de É. Picard</i>, <a href="/facts/Paris/DrNHpwnA">Paris</a>: <a href="/facts/Hermann_(publishing_house)/KjStxLLh">A. Hermann et Fils</a>, 1912. VII + 152 pp.</li> <li><a href="https://www.encyclopediaofmath.org/index.php?title=Volterra_equation">"Volterra equation"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/VolterraIntegralEquationoftheFirstKind.html">"Volterra Integral Equation of the First Kind"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/VolterraIntegralEquationoftheSecondKind.html">"Volterra Integral Equation of the Second Kind"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://eqworld.ipmnet.ru/en/solutions/ie.htm">Integral Equations: Exact Solutions</a> at EqWorld: The World of Mathematical Equations</li> <li>Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). <a href="http://apps.nrbook.com/empanel/index.html#pg=992">"Section 19.2. Volterra Equations"</a>. <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.). New York: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-88068-8.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://inteq.readthedocs.io/en/latest/">IntEQ: a Python package for numerically solving Volterra integral equations</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Polyanin, Andrei D.; Manzhirov, Alexander V. (2008). Handbook of Integral Equations (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC. ISBN 978-1584885078. <a href="978-1584885078" target="_blank">978-1584885078</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Inaba, Hisashi (2017). "The Stable Population Model". Age-Structured Population Dynamics in Demography and Epidemiology. Singapore: Springer. pp. 1–74. doi:10.1007/978-981-10-0188-8_1. ISBN 978-981-10-0187-1. <a href="978-981-10-0187-1" target="_blank">978-981-10-0187-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Brunner, Hermann (2017). Volterra Integral Equations: An Introduction to Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, UK: Cambridge University Press. ISBN 978-1107098725. <a href="978-1107098725" target="_blank">978-1107098725</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Daddi-Moussa-Ider, A.; Vilfan, A.; Golestanian, R. (6 April 2022). "Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface". Journal of Fluid Mechanics. 940: A12. arXiv:2109.14437. doi:10.1017/jfm.2022.232. <a href="/wiki/Ramin_Golestanian" target="_blank">/wiki/Ramin_Golestanian</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Daddi-Moussa-Ider, A.; Lisicki, M.; Löwen, H.; Menzel, A. M. (5 February 2020). "Dynamics of a microswimmer–microplatelet composite". Physics of Fluids. 32 (2): 021902. arXiv:2001.06646. doi:10.1063/1.5142054. <a href="/wiki/Hartmut_L%C3%B6wen" target="_blank">/wiki/Hartmut_L%C3%B6wen</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Lecture Notes on Risk Theory" (PDF). School of Mathematics, Statistics and Actuarial Science. University of Kent. February 20, 2010. pp. 17–22. <a href="https://www.kent.ac.uk/smsas/personal/lb209/files/risk-notes-10.pdf" target="_blank">https://www.kent.ac.uk/smsas/personal/lb209/files/risk-notes-10.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>