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İkinci Derece Sınır Değer Problemlerinin Matlab ile Çözümü

21 Aralık 2022 Çarşamba - # # #

Ödev

22831004155 - Hasan ŞENER
Soru 1
n = 100 alarak çözünüz.
clear
clc
warning off
close all
a = 0; b = 1; n = 100;
h = (b-a)/n;
%x'i olustur
x = [a:h:b];
% x = [a:h:b];
y = sym('y',[1 size(x,2)]);
y(1) = 1;
y(end) = -3;
% numerical differential
for j = 1:(n-1)
equation(j) = (y(j+2) - 2*y(j+1) + y(j))/(h^2) == x(j+1) - 2*y(j+1) + ((y(j+2)-y(j))/(2*h));
end
[A,B] = equationsToMatrix(equation);
Y = linsolve(A,B);
Y = vpa(Y);
Y = round(Y,3);
for t = 1:size(Y)
fprintf("| y_%d: %2.3f \t",t,Y(t))
if mod(t,9) == 0
fprintf(" \n")
end
end
| y_1: 0.759 | y_2: -2.957 | y_3: 0.730 | y_4: 0.700 | y_5: 0.671 | y_6: 0.640 | y_7: 0.610 | y_8: 0.579 | y_9: 0.547
| y_10: 0.515 | y_11: 0.483 | y_12: 0.975 | y_13: 0.450 | y_14: 0.417 | y_15: 0.384 | y_16: 0.350 | y_17: 0.315 | y_18: 0.281
| y_19: 0.246 | y_20: 0.210 | y_21: 0.174 | y_22: 0.138 | y_23: 0.950 | y_24: 0.102 | y_25: 0.065 | y_26: 0.027 | y_27: -0.010
| y_28: -0.048 | y_29: -0.086 | y_30: -0.125 | y_31: -0.164 | y_32: -0.203 | y_33: -0.243 | y_34: 0.924 | y_35: -0.283 | y_36: -0.323
| y_37: -0.363 | y_38: -0.404 | y_39: -0.445 | y_40: -0.487 | y_41: -0.528 | y_42: -0.570 | y_43: -0.612 | y_44: -0.655 | y_45: 0.897
| y_46: -0.697 | y_47: -0.740 | y_48: -0.783 | y_49: -0.827 | y_50: -0.870 | y_51: -0.914 | y_52: -0.958 | y_53: -1.002 | y_54: -1.047
| y_55: -1.091 | y_56: 0.870 | y_57: -1.136 | y_58: -1.181 | y_59: -1.226 | y_60: -1.271 | y_61: -1.316 | y_62: -1.362 | y_63: -1.407
| y_64: -1.453 | y_65: -1.499 | y_66: -1.545 | y_67: 0.843 | y_68: -1.591 | y_69: -1.637 | y_70: -1.683 | y_71: -1.729 | y_72: -1.775
| y_73: -1.821 | y_74: -1.867 | y_75: -1.914 | y_76: -1.960 | y_77: -2.006 | y_78: 0.816 | y_79: -2.052 | y_80: -2.099 | y_81: -2.145
| y_82: -2.191 | y_83: -2.237 | y_84: -2.283 | y_85: -2.329 | y_86: -2.375 | y_87: -2.420 | y_88: -2.466 | y_89: 0.787 | y_90: -2.511
| y_91: -2.557 | y_92: -2.602 | y_93: -2.647 | y_94: -2.692 | y_95: -2.737 | y_96: -2.781 | y_97: -2.825 | y_98: -2.869 | y_99: -2.913
YY = [y(1);Y;y(end)];
plot(x',YY)
grid on
Soru 2
n = 100 alarak çözünüz
clear
clc
warning off
close all
a = 1; b = 2; n =100;
h = (b-a)/n;
x = [a:h:b];
y = sym('y',[1 size(x,2)]);
y(1) = -2;
y(end) = 4;
% numerical differential
for j = 1:(n-1)
equation(j) = x(j+1)^2 - 4*y(j+1) + 4*((y(j+2)-y(j))/(2*h));
end
[A,B] = equationsToMatrix(equation);
Y = linsolve(A,B);
Y = vpa(Y);
Y = round(Y,3);
for t = 1:size(Y)
fprintf("| y_%d: %2.3f \t",t,Y(t))
if mod(t,9) == 0
fprintf(" \n")
end
end
| y_1: 6.615 | y_2: 7.205 | y_3: -1.358 | y_4: 6.582 | y_5: -1.233 | y_6: 6.551 | y_7: -1.108 | y_8: 6.523 | y_9: -0.985
| y_10: 6.496 | y_11: -0.861 | y_12: 6.772 | y_13: 6.472 | y_14: -0.739 | y_15: 6.450 | y_16: -0.617 | y_17: 6.430 | y_18: -0.496
| y_19: 6.413 | y_20: -0.376 | y_21: 6.397 | y_22: -0.256 | y_23: -1.870 | y_24: 6.384 | y_25: -0.137 | y_26: 6.373 | y_27: -0.018
| y_28: 6.364 | y_29: 0.101 | y_30: 6.357 | y_31: 0.219 | y_32: 6.352 | y_33: 0.336 | y_34: 6.729 | y_35: 6.349 | y_36: 0.454
| y_37: 6.348 | y_38: 0.571 | y_39: 6.350 | y_40: 0.687 | y_41: 6.353 | y_42: 0.804 | y_43: 6.358 | y_44: 0.920 | y_45: -1.740
| y_46: 6.366 | y_47: 1.037 | y_48: 6.375 | y_49: 1.153 | y_50: 6.387 | y_51: 1.269 | y_52: 6.400 | y_53: 1.385 | y_54: 6.416
| y_55: 1.501 | y_56: 6.689 | y_57: 6.433 | y_58: 1.617 | y_59: 6.453 | y_60: 1.733 | y_61: 6.474 | y_62: 1.849 | y_63: 6.498
| y_64: 1.965 | y_65: 6.523 | y_66: 2.082 | y_67: -1.612 | y_68: 6.551 | y_69: 2.199 | y_70: 6.581 | y_71: 2.316 | y_72: 6.612
| y_73: 2.433 | y_74: 6.646 | y_75: 2.550 | y_76: 6.681 | y_77: 2.668 | y_78: 6.651 | y_79: 6.719 | y_80: 2.787 | y_81: 6.758
| y_82: 2.906 | y_83: 6.800 | y_84: 3.025 | y_85: 6.843 | y_86: 3.145 | y_87: 6.889 | y_88: 3.265 | y_89: -1.485 | y_90: 6.936
| y_91: 3.386 | y_92: 6.986 | y_93: 3.507 | y_94: 7.038 | y_95: 3.629 | y_96: 7.092 | y_97: 3.752 | y_98: 7.147 | y_99: 3.876
YY = [y(1);Y;y(end)];
plot(x',YY)
grid on



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