APPOROXIMATE SOLUTION OF LAPLACE'S EQUATION

PLEASE I HAVE THIS QUESTION:

APPROXIMATE THE SOLUTION OF LAPLACE'S EQUATION IN THE SQUARE ABCD FOR THE BOUNDARY CONDITIONS INDICATED BELOW WITH THE SPACING h=1/6 for the following values of the parameters.

α=0.9+0.1,k; k=0,1,2; β=1.01+0.02,n; n=0,1,2;
Carry out Iterations to within 10-²

J=  1  2  3  4  5  6
u/AD  0  17.28  29.05α  40.00  29.05β  17.28
u/BC  0  9.81  17.98 α  29.12  38.25 β  42.31
u/AB  0  0  0  0  0  0
u/DC  4.31  6.98  12.38 β  19.14  30.10 α  40.16

CAN YOU PLEASE HELP ME ON A PROGRAM TO GET IT SOLVED USING GAUSS SEIDEL I NEED TO SUBMIT IT IN 3 DAYS TIME, THANKS THESE ARE THE EQUATION I USED MANUALLY:

L1=0.25*(0+L6+L2+17.28)

L2=0.25*(L1+L7+L3+29.05α

L3=0.25*(L2+L8+L4+40)

L4=0.25*(L3+L9+L5+29.05β

L5=0.25*(L4+L10+6.98+17.28)

L6=0.25*(0+L11+L7+L1)

L7=0.25*(L6+L12+L8+L12)

L8=0.25*(L7+L13+L9+L3)

L9=0.25*(L8+L14+L10+L4)

L10=0.25*(L9+L15+12.38β+L5)

L11=0.25*(0+L16+L12+L6)

L12=0.25*(L11+L17+L13+L7)

L13=0.25*(L12+L18+L14+L8)

L14=0.25*(L13+L19+L15+L9)

L15=0.25*(L14+L20+19.14+L10)

L16=0.25*(0+L21+L17+L11)

L17=0.25*(L16+L22+L18+L12)

L18=0.25*(L17+L23+L19+L13)

L19=0.25*(L18+L24+L20+L14)

L20=0.25*(L19+L25+30.10α+L15)

L21=0.25*(0+0+L22+L16)

L22=0.25*(L21+9.81+L23+L17)

L23=0.25*(L22+17.98α+L24+L18)

L24=0.25*(L23+29.12+L25+L19)

L25=0.25*(L24+38.25β+40.16+L20)

[Walter Roberson]

Which equation are you trying to solve? In written form, not by name.

[Anie Ekpes]

I AM TRYING TO SOLVE THE PROBLEM USING GAUSS SEIDEL METHOD, I HAVE DONE IT MANUALLY BUT I NEED TO ITERATE IT FOR 10 TIMES AND THE MESH POINTS ARE 25, MEANING I HAVE TO SOLVE 250 EQUATIONS.

THIS IS THE EQUATION I USED MANUALLY:

L1=0.25*(0+L6+L2+17.28)

L2=0.25*(L1+L7+L3+29.05α

L3=0.25*(L2+L8+L4+40)

L4=0.25*(L3+L9+L5+29.05β

L5=0.25*(L4+L10+6.98+17.28)

L6=0.25*(0+L11+L7+L1)

L7=0.25*(L6+L12+L8+L12)

L8=0.25*(L7+L13+L9+L3)

L9=0.25*(L8+L14+L10+L4)

L10=0.25*(L9+L15+12.38β+L5)

L11=0.25*(0+L16+L12+L6)

L12=0.25*(L11+L17+L13+L7)

L13=0.25*(L12+L18+L14+L8)

L14=0.25*(L13+L19+L15+L9)

L15=0.25*(L14+L20+19.14+L10)

L16=0.25*(0+L21+L17+L11)

L17=0.25*(L16+L22+L18+L12)

L18=0.25*(L17+L23+L19+L13)

L19=0.25*(L18+L24+L20+L14)

L20=0.25*(L19+L25+30.10α+L15)

L21=0.25*(0+0+L22+L16)

L22=0.25*(L21+9.81+L23+L17)

L23=0.25*(L22+17.98α+L24+L18)

L24=0.25*(L23+29.12+L25+L19)

L25=0.25*(L24+38.25β+40.16+L20)

Answer 1

[nick]

Hello Anie,

To approximate the solution of Laplace's equation using the Gauss-Seidel method in MATLAB, you can follow the steps below :

    % Parameters
    h = 1/6;
    tol = 1e-2; 
    maxIter = 1000; 

    % Boundary conditions
    uAD = [0, 17.28, 29.05, 40.00, 29.05, 17.28];
    uBC = [0, 9.81, 17.98, 29.12, 38.25, 42.31];
    uAB = [0, 0, 0, 0, 0, 0];
    uDC = [4.31, 6.98, 12.38, 19.14, 30.10, 40.16];

    % Initialize grid
    L = zeros(5, 5);
    alpha_values = 0.9 + 0.1 * (0:2);
    beta_values = 1.01 + 0.02 * (0:2);

    % Iterate over alpha and beta values
    for alpha = alpha_values
        for beta = beta_values
            % Adjust boundary conditions with alpha and beta
            uAD(3) = 29.05 * alpha;
            uAD(5) = 29.05 * beta;
            uBC(3) = 17.98 * alpha;
            uBC(5) = 38.25 * beta;
            uDC(3) = 12.38 * beta;
            uDC(5) = 30.10 * alpha;

            % Gauss-Seidel iteration
            for iter = 1:maxIter
                L_old = L;
     
                for i = 1:5
                    for j = 1:5
                        if i == 1
                            top = uAB(j + 1);
                        else
                            top = L(i - 1, j);
                        end
                        if i == 5
                            bottom = uDC(j + 1);
                        else
                            bottom = L(i + 1, j);
                        end
                        if j == 1
                            left = uAD(i + 1);
                        else
                            left = L(i, j - 1);
                        end
                        if j == 5
                            right = uBC(i + 1);
                        else
                            right = L(i, j + 1);
                        end

                        L(i, j) = 0.25 * (top + bottom + left + right);
                    end
                end

            end

         
            fprintf('Results for alpha=%.2f, beta=%.2f:\n', alpha, beta);
            disp(L);
        end
    end

Results for alpha=0.90, beta=1.01:

   11.4605    8.7489    7.6987    7.6721    8.3259
   19.8133   15.8362   14.3740   14.6638   15.8215
   25.8113   20.4089   19.2971   20.7876   24.1144
   23.0229   20.6908   21.6181   25.0751   30.7283
   16.2490   17.7133   21.4095   27.1664   35.0912

Results for alpha=0.90, beta=1.03:

   11.4767    8.7732    7.7255    7.6973    8.3434
   19.8536   15.8907   14.4314   14.7203   15.8662
   25.9021   20.5045   19.3891   20.8864   24.2190
   23.2503   20.8361   21.7340   25.2172   31.0033
   16.3415   17.8557   21.4937   27.2451   35.1796

Results for alpha=0.90, beta=1.05:

   11.4929    8.7976    7.7522    7.7225    8.3608
   19.8940   15.9451   14.4888   14.7768   15.9108
   25.9930   20.6001   19.4810   20.9852   24.3236
   23.4777   20.9815   21.8500   25.3593   31.2784
   16.4340   17.9981   21.5780   27.3238   35.2681

Results for alpha=1.00, beta=1.01:

   11.7935    9.0274    7.9320    7.9050    8.5639
   20.8665   16.3841   14.7956   15.1243   16.5405
   26.2384   20.8469   19.7421   21.2558   24.4940
   23.2401   21.0229   22.0702   25.6629   31.0596
   16.3587   17.9345   21.8527   28.2661   35.4489

Results for alpha=1.00, beta=1.03:

   11.8096    9.0517    7.9587    7.9302    8.5813
   20.9069   16.4385   14.8530   15.1808   16.5852
   26.3292   20.9425   19.8341   21.3546   24.5986
   23.4675   21.1683   22.1861   25.8050   31.3346
   16.4511   18.0769   21.9370   28.3448   35.5374

Results for alpha=1.00, beta=1.05:

   11.8258    9.0761    7.9855    7.9554    8.5988
   20.9472   16.4930   14.9104   15.2373   16.6298
   26.4201   21.0382   19.9260   21.4534   24.7032
   23.6949   21.3136   22.3020   25.9472   31.6097
   16.5436   18.2194   22.0212   28.4235   35.6258

Results for alpha=1.10, beta=1.01:

   12.1264    9.3059    8.1653    8.1380    8.8019
   21.9197   16.9319   15.2173   15.5847   17.2596
   26.6655   21.2849   20.1871   21.7240   24.8736
   23.4573   21.3551   22.5222   26.2507   31.3909
   16.4683   18.1558   22.2960   29.3658   35.8067

Results for alpha=1.10, beta=1.03:

   12.1426    9.3302    8.1920    8.1631    8.8193
   21.9601   16.9864   15.2747   15.6412   17.3042
   26.7563   21.3805   20.2790   21.8228   24.9782
   23.6847   21.5004   22.6381   26.3929   31.6659
   16.5607   18.2982   22.3802   29.4445   35.8951

Results for alpha=1.10, beta=1.05:

   12.1588    9.3546    8.2187    8.1883    8.8368
   22.0004   17.0408   15.3321   15.6977   17.3488
   26.8472   21.4762   20.3710   21.9216   25.0829
   23.9122   21.6457   22.7540   26.5350   31.9410
   16.6532   18.4406   22.4645   29.5233   35.9836

You can refer to the documentation by executing the following MATLAB Command Window to know more about MATLAB language fundamentals:

doc language-fundamentals