Abstract
Volterra studied the hereditary influences when he was examining a population growth model. The research work resulted in a specific topic, where both differential and integral operators appeared together in the same equation. This new type of equations was termed as Volterra integro-differential equations [1–4], given in the form
Where \({u^{\left( n \right)}}\left( x \right) = \frac{{{d^n}u}}{{d{x^n}}}\). Because the resulted equation in (5.1) combines the differential operator and the integral operator, then it is necessary to define initial conditions u(0), u′ (0), , u (n−1)(0) for the determination of the particular solution u(x) of the Volterra integro-differential equation (5.1). Any Volterra integro-differential equation is characterized by the existence of one or more of the derivatives u′ (x), u″ (x), outside the integral sign. The Volterra integro-differential equations may be observed when we convert an initial value problem to an integral equation by using Leibnitz rule.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L.G. Chambers, Integral Equations, A Short Course, International Textbook Company, London, (1976).
R.F. Churchhouse, Handbook of Applicable Mathematics, Wiley, New York, (1981).
B.L. Moiseiwitsch, Integral Equations, Longman, London and New York, (1977).
V. Volterra, Theory of Functionals of Integral and Integro-Differential Equations, Dover, New York, (1959).
G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, (1986).
G. Adomian, Solving Frontier Problems of Physics, The Decomposition Method, Kluwer, Boston, (1994).
G. Adomian and R. Rach, Noise terms in decomposition series solution, Comput. Math. Appl., 24 (1992) 61–64.
A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin, (2009).
A.M. Wazwaz, A First Course in Integral Equations, World Scientific, Singapore, (1997).
A.M. Wazwaz, The variational iteration method; a reliable tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54 (2007) 926–932.
P. Linz, A simple approximation method for solving Volterra integro-differential equations of the first kind, J. Inst. Math. Appl., 14 (1974) 211–215.
P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, (1985).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Wazwaz, AM. (2011). Volterra Integro-Differential Equations. In: Linear and Nonlinear Integral Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21449-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-21449-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21448-6
Online ISBN: 978-3-642-21449-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)