I tried to compute this exact solution, but faced difficulty if the value of [itex] η[/itex] approaching to [itex] ζ [/itex]. Let say the value of [itex] ζ [/itex] is fix at 0.5 and the collocation points for η is from 0 to 1.(adsbygoogle = window.adsbygoogle || []).push({});

[tex] θ(η,ζ)=e^{-ε\frac{η}{2}} \left\{ e^{-η}+\left(1-\frac{ε^2}{4}\right)^{1/2} η \int_η^ζ e^{-τ}\frac{I_1 \left\{\left[ \left(1-\frac{ε^2}{4}\right)\left(τ^2-η^2\right) \right]^{1/2} \right\}} {\left(τ^2-η^2\right)} \right\} U(ζ-η)[/tex]

These are the values that is suppose to appear, but only when η=0.5 θ=0.295778, i don't manage to get that value, others is ok. I used trapz command in matlab to calculate the area.

η=0.0 θ=1.000000

η=0.1 θ=0.915287

η=0.2 θ=0.831763

η=0.3 θ=0.749758

η=0.4 θ=0.669587

η=0.5 θ=0.295778

η=0.6 θ=0.000000

η=0.7 θ=0.000000

η=0.8 θ=0.000000

η=0.9 θ=0.000000

η=1.0 θ=0.000000

I do suspect that the integration of Bessel function is not simply become 0 when η=0.5 (approach to singularity to that point). I do appreciate if someone could give some advice.

Here I attach the matlab program that I wrote. Thank you in advance

format short

%analytic solution

tic

ita=0:0.1:1; m=11;

ep=0.1;

zeta=0.5;

area=zeros(1,m);

%kira integration dahulu

for i=1:m

if ita(i)<=zeta

tau=linspace(ita(i)+0.000001,0.5,100000);

%argument for bessel function

a=(1-(ep^2)/4);b=(tau.^2-ita(i)^2);

Z=(a*b).^(1/2);

%Modified bessel function

func=@(tau) (exp(-tau).*besseli(1,Z))./sqrt(b);

area(i)=trapz(tau,func(tau));

else

area(i)=0;

end

end

Theta=zeros(1,m);

for i=1:m

if ita(i)<=zeta

Theta(i)=exp((-ita(i)./2)*ep)*(exp(-ita(i))+sqrt(a)*ita(i).*area(i));

else

Theta(i)=0;

end

end

plot(ita,Theta);

axis([0 2.2 0 1]);

tableresult(:,1)=ita';

tableresult(:,2)=Theta';

disp(' x Analytic')

disp('')

disp(tableresult);

toc

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# Computing Integration of Bessel Function

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