Solve differential equation where parameter is dependent on value of an integrated variable

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Jake Swanson 2019-12-6

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https://ww2.mathworks.cn/matlabcentral/answers/495284-solve-differential-equation-where-parameter-is-dependent-on-value-of-an-integrated-variable

评论： darova 2019-12-11

Hi,

I have a system of differental equations where I have a parameter kw(lambda), which is the absorption spectrum of water at a specific wavelength, lambda. For kw I have a data table with wavelengths from 400 to 700 by increments of 2.5 (400, 402.5, 405...) with the corresponding absorption value at each wavelength. I have the wavelengths stored as a parameter wave and the absorption values as a parameter abs.

In the system of equations I have to integrate over lambda from 400 to 700; therefore the value of lambda used to look up kw is determined by the integral. I want to interpolate the value of kw with:

kw=interp1(wave,abs,lambda)

I've tried to define kw as above and then solve my equations but get the error "Unrecognized function or variable 'lambda' ". Is there a way to interpolate over a set of values that are determined by an integral or is there another way that I should be going about solving this problem?

Thanks,

Jake

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Jake Swanson 2019-12-6

在 MATLAB Online 中打开

function j = spectrum(t,N) 
    
    pmax= 100;
    phi= 2;
    L= 0.1;
    a= 100;
    k= 105;
    Iin=100;
    wave=[380;382.500000000000;385;387.500000000000;390;392.500000000000;395;397.500000000000;400;402.500000000000;405;407.500000000000;410;412.500000000000;415;417.500000000000;420;422.500000000000;425;427.500000000000;430;432.500000000000;435;437.500000000000;440;442.500000000000;445;447.500000000000;450;452.500000000000;455;457.500000000000;460;462.500000000000;465;467.500000000000;470;472.500000000000;475;477.500000000000;480;482.500000000000;485;487.500000000000;490;492.500000000000;495;497.500000000000;500;502.500000000000;505;507.500000000000;510;512.500000000000;515;517.500000000000;520;522.500000000000;525;527.500000000000;530;532.500000000000;535;537.500000000000;540;542.500000000000;545;547.500000000000;550;552.500000000000;555;557.500000000000;560;562.500000000000;565;567.500000000000;570;572.500000000000;575;577.500000000000;580;582.500000000000;585;587.500000000000;590;592.500000000000;595;597.500000000000;600;602.500000000000;605;607.500000000000;610;612.500000000000;615;617.500000000000;620;622.500000000000;625;627.500000000000;630;632.500000000000;635;637.500000000000;640;642.500000000000;645;647.500000000000;650;652.500000000000;655;657.500000000000;660;662.500000000000;665;667.500000000000;670;672.500000000000;675;677.500000000000;680;682.500000000000;685;687.500000000000;690;692.500000000000;695;697.500000000000;700;702.500000000000;705;707.500000000000;710;712.500000000000;715;717.500000000000;720;722.500000000000;725;727.500000000000];
    abs=[0.0113700000000000;0.0104400000000000;0.00941000000000000;0.00917000000000000;0.00851000000000000;0.00829000000000000;0.00813000000000000;0.00775000000000000;0.00663000000000000;0.00579000000000000;0.00530000000000000;0.00503000000000000;0.00473000000000000;0.00452000000000000;0.00444000000000000;0.00442000000000000;0.00454000000000000;0.00474000000000000;0.00478000000000000;0.00482000000000000;0.00495000000000000;0.00504000000000000;0.00530000000000000;0.00580000000000000;0.00635000000000000;0.00696000000000000;0.00751000000000000;0.00830000000000000;0.00922000000000000;0.00969000000000000;0.00962000000000000;0.00957000000000000;0.00979000000000000;0.0100500000000000;0.0101100000000000;0.0102000000000000;0.0106000000000000;0.0109000000000000;0.0114000000000000;0.0121000000000000;0.0127000000000000;0.0131000000000000;0.0136000000000000;0.0144000000000000;0.0150000000000000;0.0162000000000000;0.0173000000000000;0.0191000000000000;0.0204000000000000;0.0228000000000000;0.0256000000000000;0.0280000000000000;0.0325000000000000;0.0372000000000000;0.0396000000000000;0.0399000000000000;0.0409000000000000;0.0416000000000000;0.0417000000000000;0.0428000000000000;0.0434000000000000;0.0447000000000000;0.0452000000000000;0.0466000000000000;0.0474000000000000;0.0489000000000000;0.0511000000000000;0.0537000000000000;0.0565000000000000;0.0593000000000000;0.0596000000000000;0.0606000000000000;0.0619000000000000;0.0640000000000000;0.0642000000000000;0.0672000000000000;0.0695000000000000;0.0733000000000000;0.0772000000000000;0.0836000000000000;0.0896000000000000;0.0989000000000000;0.110000000000000;0.122000000000000;0.135100000000000;0.151600000000000;0.167200000000000;0.192500000000000;0.222400000000000;0.247000000000000;0.257700000000000;0.262900000000000;0.264400000000000;0.266500000000000;0.267800000000000;0.270700000000000;0.275500000000000;0.281000000000000;0.283400000000000;0.290400000000000;0.291600000000000;0.299500000000000;0.301200000000000;0.307700000000000;0.310800000000000;0.322000000000000;0.325000000000000;0.335000000000000;0.340000000000000;0.358000000000000;0.371000000000000;0.393000000000000;0.410000000000000;0.424000000000000;0.429000000000000;0.436000000000000;0.439000000000000;0.448000000000000;0.448000000000000;0.461000000000000;0.465000000000000;0.478000000000000;0.486000000000000;0.502000000000000;0.516000000000000;0.538000000000000;0.559000000000000;0.592000000000000;0.624000000000000;0.663000000000000;0.704000000000000;0.756000000000000;0.827000000000000;0.914000000000000;1.00700000000000;1.11900000000000;1.23100000000000;1.35600000000000;1.48900000000000;1.67800000000000];
    Kw= interp1(wave,abs,lambda);  
    s= 0.017;
    zm= 10;  
    K=1;
     
        
        j= (N/zm)* integral(@growth2,0,10,'ArrayValued',true);
            
      
          function j = growth2(lambda)
              
        j = integral (@photons,400,700,'ArrayValued',true);
        j= ((pmax * j)/((pmax/phi) + j)) - L * N;
                         
 
              function j=photons(z)
        
        kbg =K * exp(-(s)*(lambda - 484));    
        I= Iin * exp((-Kw*z) -(kbg*z) -sum(k*N*z)); 
        j=a*k*I;
     
        end
     end
end