Volterra-Fredholm Integro-Differential Equations

Abdul-Majid Wazwaz ()
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Abdul-Majid Wazwaz: Saint Xavier University

Chapter Chapter 9 in Linear and Nonlinear Integral Equations, 2011, pp 285-309 from Springer

Abstract: Abstract The Volterra-Fredholm integro-differential equations [1–4] appear in two types, namely: 9.1 $${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + {\lambda _1}\int_a^x {{K_1}\left( {x,t} \right)u\left( t \right)dt + {\lambda _2}\int_a^b {{K_2}\left( {x,t} \right)u\left( t \right)dt} ,} $$ and the mixed form 9.2 $${u^{\left( k \right)}}\left( x \right) = f\left( x \right) + \lambda \int_0^x {\int_a^b {K\left( {r,t} \right)u\left( t \right)dtdr} ,} $$ where $${u^{\left( k \right)}}\left( x \right) = \frac{{{d^k}u\left( x \right)}}{{d{x^k}}}$$ . The first type contains disjoint integrals and the second type contains mixed integrals such that the Fredholm integral is the interior one, and Volterra is the exterior integral.

Date: 2011
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DOI: 10.1007/978-3-642-21449-3_9

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