Abstract
Integral equations of Volterra type arise quite naturally in physical applications modelled by initial-value problems. Consider the linear Volterra equation of the second kind. (Fredholm equations of the second kind which are associated with boundary value problems for a fmite interval [a,b], are similar except that the upper limit is b.)
with a ≤ x, y ≤ b, and let λ = 1. Using decomposition EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\varphi =\sum\nolimits_{n=0}^{\infty }{{{\varphi }_{n}}}\left( x \right)$$, we identify φ 0 = f(x) assuming f(x) ≠ 0,
then and write EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\varphi =\sum\nolimits_{n=0}^{\infty }{{{\varphi }_{n}}}$$ as the solution, or with an m-term approximant EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabgU % caRiaadkeaaaa!3864!]></EquationSource><EquationSource Format="TEX"><![CDATA[$$\varphi =\sum\nolimits_{m=0}^{m-1}{{{\varphi }_{n}}}$$. In some cases, exact solutions are determinable. Consider an example:
Then
Thus the two-term approximant is
or φ = sin x as is easily verified either by calculating more terms or by substitution.
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Reference
Y. Cherruault, G. Saccomandi, and B. Some, New Results for the Convergence of Adomian’s Method Applied to Integral Equations, Math. Comput. Modelling, 16, (8593) (1992).
Suggested Reading
G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, New York (1986).
B. Some, Some Recent Numerical Methods for Solving Hammersteins Integral Equations, Math. Comput. Modelling, to appear.
B. Some, A New Computational Method for Solving Integral Equations, submitted for publication.
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© 1994 Springer Science+Business Media Dordrecht
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Adomian, G. (1994). Integral Equations. In: Solving Frontier Problems of Physics: The Decomposition Method. Fundamental Theories of Physics, vol 60. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8289-6_10
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DOI: https://doi.org/10.1007/978-94-015-8289-6_10
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