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Semi-Markov Models by Yuriy E Obzherin, Elena G Boyko

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Appendix C

The Solution of the System of Integral Equation (3.6)

Let us prove the formula (3.9) to determine the solution of (3.6).

Introduce the operator: Arϕ(x1,x2)=0ϕ(x1+t,x2+t)r(t)dtimage, then the second equation of the system (3.6) can be rewritten as follows:

ϕ1=ρ0Arf1(x1)f2(x2)+Arϕ1+Arf1(x1)ϕ4(x2)+Arϕ5(x1)f2(x2),

image
then,

IArϕ1=ρ0Arf1(x1)f2(x2)+Arf1(x1)ϕ4(x2)+Arϕ5(x1)f2(x2),ϕ1=ρ0IAr1Arf1(x1)f2(x2)+IAr1Arf1(x1)ϕ4(x2)++IAr1Arϕ5(x1)f2(x2),(IAr)1=I+n=1Arn,Arnϕ(x1,x2)=0ϕ(x1+t,x2+t)r*(n)(t)dt,n=1Arnϕ=0ϕ(x1+t,x2+t)hr(t)dt

where

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