Решение алгебраических и дифференциальных уравнений в пакете MatLab

 

МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ

Федеральное государственное автономное образовательное учреждение

высшего профессионального образования

«Дальневосточный федеральный университет»

ИНЖЕНЕРНАЯ ШКОЛА

Кафедра электроэнергетики и электротехники

КУРСОВАЯ РАБОТА

«Решение алгебраических и дифференциальных уравнений в пакете MatLab»

Студент гр. Б-3205б

Хомячук Артём Игоревич

Руководитель доцент Н.М. Марченко

г. Владивосток 2013

1)Решение алгебраических уравнений.

1.(а) Задан один входной параметр:

Файл-функции:

f1=fun(x)=1.*cos(2.*x).*sin((pi./3)+x);

Слово function в первой строке определяет, что данный файл

содержит файл-функцию. Сохраняем этот файл с соответствующим именем fun. Далее в редакторе М-файлов пишем текст файл-программы:

Файл-программа:

fprintf('Таблица Значений Функции f1\n');

fprintf('----------------------------\n');('| x | y |\n');('----------------------------\n');=[0:0.07:7];=fun(x);=[x,y];('| %6.2f | %6.2f |\n',v)(x,y,'m*-')onon('fun',[0 7],'c-')

Результат работы программы:

Таблица Значений Функции f1

-------------------------

| x | y |

-------------------------

| 0.00 | 0.07 |

| 0.14 | 0.21 |

| 0.28 | 0.35 |

| 0.42 | 0.49 |

| 0.56 | 0.63 |

| 0.70 | 0.77 |

| 0.84 | 0.91 |

| 0.98 | 1.05 |

| 1.12 | 1.19 |

| 1.26 | 1.33 |

| 1.40 | 1.47 |

| 1.54 | 1.61 |

| 1.68 | 1.75 |

| 1.82 | 1.89 |

| 1.96 | 2.03 |

| 2.10 | 2.17 |

| 2.24 | 2.31 |

| 2.38 | 2.45 |

| 2.52 | 2.59 |

| 2.66 | 2.73 |

| 2.80 | 2.87 |

| 2.94 | 3.01 |

| 3.08 | 3.15 |

| 3.22 | 3.29 |

| 3.36 | 3.43 |

| 3.50 | 3.57 |

| 3.64 | 3.71 |

| 3.78 | 3.85 |

| 3.92 | 3.99 |

| 4.06 | 4.13 |

| 4.20 | 4.27 |

| 4.34 | 4.41 |

| 4.48 | 4.55 |

| 4.62 | 4.69 |

| 4.76 | 4.83 |

| 4.90 | 4.97 |

| 5.04 | 5.11 |

| 5.18 | 5.25 |

| 5.32 | 5.39 |

| 5.46 | 5.53 |

| 5.60 | 5.67 |

| 5.74 | 5.81 |

| 5.88 | 5.95 |

| 6.02 | 6.09 |

| 6.16 | 6.23 |

| 6.30 | 6.37 |

| 6.44 | 6.51 |

| 6.58 | 6.65 |

| 6.72 | 6.79 |

| 6.86 | 6.93 |

| 7.00 | 0.87 |

| 0.89 | 0.89 |

| 0.87 | 0.82 |

| 0.75 | 0.66 |

| 0.56 | 0.44 |

| 0.30 | 0.17 |

| 0.03 | -0.10 |

| -0.23 | -0.34 |

| -0.44 | -0.51 |

| -0.57 | -0.60 |

| -0.61 | -0.60 |

| -0.57 | -0.53 |

| -0.46 | -0.39 |

| -0.32 | -0.24 |

| -0.16 | -0.10 |

| -0.04 | 0.00 |

| 0.03 | 0.03 |

| 0.02 | -0.01 |

| -0.06 | -0.13 |

| -0.21 | -0.31 |

| -0.40 | -0.50 |

| -0.60 | -0.69 |

| -0.77 | -0.83 |

| -0.87 | -0.89 |

| -0.89 | -0.86 |

| -0.82 | -0.74 |

| -0.65 | -0.54 |

| -0.42 | -0.29 |

| -0.15 | -0.01 |

| 0.12 | 0.24 |

| 0.35 | 0.45 |

| 0.52 | 0.57 |

| 0.60 | 0.61 |

| 0.60 | 0.57 |

| 0.52 | 0.46 |

| 0.38 | 0.31 |

| 0.23 | 0.15 |

| 0.09 | 0.03 |

| -0.01 | -0.03 |

| -0.03 | -0.02 |

| 0.02 | 0.07 |

| 0.14 | 0.22 |

| 0.32 | 0.42 |

| 0.51 | 0.61 |

| 0.70 | 0.77 |

| 0.83 | 0.87 |

| 0.89 | 0.89 |

| 0.86 | 0.81 |

| 0.73 | 0.64 |

| 0.53 | 0.40 |

| 0.27 | 0.13 |

2.(б) Задано несколько входных параметров:

Файл-функция:

дифференциальный уравнение эйлер программа

function r=rad(x,y,z)

r=((x^4)+5*(y^2)+21*z)/(sqrt((x^3)+1));

Файл-программа:

=rad(1,2,3)

Результат работы программы: r = 59.3970

3.(в) Задано несколько выходных аргументов:

Файл-функция:

function [f3,f4,f5]=mad(x)=2*sin(3*x)+3*cos(2*x);=(sqrt(x+1)+2*x)/((x^2)+1);

f5=-4*(x^(-2));

Файл-программа:

x=7;

[f3,f4,f5]=mad(x)

Результат работы программы:

f3 = 2.0835= 0.3366= -0.0816

Решение дифференциальных уравнений

,

а) с использованием классических алгоритмов численных методов Эйлера и Рунге-Кутта4-го порядка; Метод Эйлера:

Алгоритм решения дифференциального уравнения методом Эйлера

Файл-функция:

function f=fun(x)=x;

Файл-программа:

t0=0;=5;=0.05;=5;=10;=0;('Таблица значений по МЭ\n');('..........................\n');('| t | x | y |\n');

fprintf('..........................\n');t=t0:h:tk

x=(a/b)*sqrt(t);

f=faaan(x);

y=y+h*f;

fprintf('| %4.2f | %4.4f | %4.4f |\n',t,x,y);

plot(t,x,'r+')

grid on

hold on

plot(t,y,'b*')

xlabel('t');

ylabel('y');

Результат работы программы:

Таблица значений по МЭ

..........................

| t | x | y _ |

......................................

| 0.00 | 0.0000 | 0.0000 |

| 0.01 | 0.0500 | 0.0005 |

| 0.02 | 0.0707 | 0.0012 |

| 0.03 | 0.0866 | 0.0021 |

| 0.04 | 0.1000 | 0.0031 |

| 0.05 | 0.1118 | 0.0042 |

| 0.06 | 0.1225 | 0.0054 |

| 0.07 | 0.1323 | 0.0067 |

| 0.08 | 0.1414 | 0.0082 |

| 0.09 | 0.1500 | 0.0097 |

| 0.10 | 0.1581 | 0.0112 |

| 0.11 | 0.1658 | 0.0129 |

| 0.12 | 0.1732 | 0.0146 |

| 0.13 | 0.1803 | 0.0164 |

| 0.14 | 0.1871 | 0.0183 |

| 0.15 | 0.1936 | 0.0202 |

| 0.16 | 0.2000 | 0.0222 |

| 0.17 | 0.2062 | 0.0243 |

| 0.18 | 0.2121 | 0.0264 |

| 0.19 | 0.2179 | 0.0286 |

| 0.20 | 0.2236 | 0.0308 |

| 0.21 | 0.2291 | 0.0331 |

| 0.22 | 0.2345 | 0.0355 |

| 0.23 | 0.2398 | 0.0379 |

| 0.24 | 0.2449 | 0.0403 |

| 0.25 | 0.2500 | 0.0428 |

| 0.26 | 0.2550 | 0.0454 |

| 0.27 | 0.2598 | 0.0480 |

| 0.28 | 0.2646 | 0.0506 |

| 0.29 | 0.2693 | 0.0533 |

| 0.30 | 0.2739 | 0.0560 |

| 0.31 | 0.2784 | 0.0588 |

| 0.32 | 0.2828 | 0.0617 |

| 0.33 | 0.2872 | 0.0645 |

| 0.34 | 0.2915 | 0.0674 |

| 0.35 | 0.2958 | 0.0704 |

| 0.36 | 0.3000 | 0.0734 |

| 0.37 | 0.3041 | 0.0764 |

| 0.38 | 0.3082 | 0.0795 |

| 0.39 | 0.3122 | 0.0826 |

| 0.40 | 0.3162 | 0.0858 |

| 0.41 | 0.3202 | 0.0890 |

| 0.42 | 0.3240 | 0.0922 |

| 0.43 | 0.3279 | 0.0955 |

| 0.44 | 0.3317 | 0.0988 |

| 0.45 | 0.3354 | 0.1022 |

| 0.46 | 0.3391 | 0.1056 |

| 0.47 | 0.3428 | 0.1090 |

| 0.48 | 0.3464 | 0.1125 |

| 0.49 | 0.3500 | 0.1160 |

| 0.50 | 0.3536 | 0.1195 |

| 0.51 | 0.3571 | 0.1231 |

| 0.52 | 0.3606 | 0.1267 |

| 0.53 | 0.3640 | 0.1303 |

| 0.54 | 0.3674 | 0.1340 |

| 0.55 | 0.3708 | 0.1377 |

| 0.56 | 0.3742 | 0.1415 |

| 0.57 | 0.3775 | 0.1452 |

| 0.58 | 0.3808 | 0.1490 |

| 0.59 | 0.3841 | 0.1529 |

| 0.60 | 0.3873 | 0.1568 |

| 0.61 | 0.3905 | 0.1607 |

| 0.62 | 0.3937 | 0.1646 |

| 0.63 | 0.3969 | 0.1686 |

| 0.64 | 0.4000 | 0.1726 |

| 0.65 | 0.4031 | 0.1766 |

| 0.66 | 0.4062 | 0.1807 |

| 0.67 | 0.4093 | 0.1848 |

| 0.68 | 0.4123 | 0.1889 |

| 0.69 | 0.4153 | 0.1930 |

| 0.70 | 0.4183 | 0.1972 |

| 0.71 | 0.4213 | 0.2014 |

| 0.72 | 0.4243 | 0.2057 |

| 0.73 | 0.4272 | 0.2099 |

| 0.74 | 0.4301 | 0.2142 |

| 0.75 | 0.4330 | 0.2186 |

| 0.76 | 0.4359 | 0.2229 |

| 0.77 | 0.4387 | 0.2273 |

| 0.78 | 0.4416 | 0.2317 |

| 0.79 | 0.4444 | 0.2362 |

| 0.80 | 0.4472 | 0.2406 |

| 0.81 | 0.4500 | 0.2451 |

| 0.82 | 0.4528 | 0.2497 |

| 0.83 | 0.4555 | 0.2542 |

| 0.84 | 0.4583 | 0.2588 |

| 0.85 | 0.4610 | 0.2634 |

| 0.86 | 0.4637 | 0.2681 |

| 0.87 | 0.4664 | 0.2727 |

| 0.88 | 0.4690 | 0.2774 |

| 0.89 | 0.4717 | 0.2821 |

| 0.90 | 0.4743 | 0.2869 |

| 0.91 | 0.4770 | 0.2916 |

| 0.92 | 0.4796 | 0.2964 |

| 0.93 | 0.4822 | 0.3013 |

| 0.94 | 0.4848 | 0.3061 |

| 0.95 | 0.4873 | 0.3110 |

| 0.96 | 0.4899 | 0.3159 |

| 0.97 | 0.4924 | 0.3208 |

| 0.98 | 0.4950 | 0.3258 |

| 0.99 | 0.4975 | 0.3307 |

| 1.00 | 0.5000 | 0.3357 |

| 1.01 | 0.5025 | 0.3408 |

| 1.02 | 0.5050 | 0.3458 |

| 1.03 | 0.5074 | 0.3509 |

| 1.04 | 0.5099 | 0.3560 |

| 1.05 | 0.5123 | 0.3611 |

| 1.06 | 0.5148 | 0.3663 |

| 1.07 | 0.5172 | 0.3714 |

| 1.08 | 0.5196 | 0.3766 |

| 1.09 | 0.5220 | 0.3818 |

| 1.10 | 0.5244 | 0.3871 |

| 1.11 | 0.5268 | 0.3924 |

| 1.12 | 0.5292 | 0.3976 |

| 1.13 | 0.5315 | 0.4030 |

| 1.14 | 0.5339 | 0.4083 |

| 1.15 | 0.5362 | 0.4137 |

| 1.16 | 0.5385 | 0.4190 |

| 1.17 | 0.5408 | 0.4245 |

| 1.18 | 0.5431 | 0.4299 |

| 1.19 | 0.5454 | 0.4353 |

| 1.20 | 0.5477 | 0.4408 |

| 1.21 | 0.5500 | 0.4463 |

| 1.22 | 0.5523 | 0.4518 |

| 1.23 | 0.5545 | 0.4574 |

| 1.24 | 0.5568 | 0.4630 |

| 1.25 | 0.5590 | 0.4685 |

| 1.26 | 0.5612 | 0.4742 |

| 1.27 | 0.5635 | 0.4798 |

| 1.28 | 0.5657 | 0.4854 |

| 1.29 | 0.5679 | 0.4911 |

| 1.30 | 0.5701 | 0.4968 |

| 1.31 | 0.5723 | 0.5025 |

| 1.32 | 0.5745 | 0.5083 |

| 1.33 | 0.5766 | 0.5141 |

| 1.34 | 0.5788 | 0.5198 |

| 1.35 | 0.5809 | 0.5257 |

| 1.36 | 0.5831 | 0.5315 |

| 1.37 | 0.5852 | 0.5373 |

| 1.38 | 0.5874 | 0.5432 |

| 1.39 | 0.5895 | 0.5491 |

| 1.40 | 0.5916 | 0.5550 |

| 1.41 | 0.5937 | 0.5610 |

| 1.42 | 0.5958 | 0.5669 |

| 1.43 | 0.5979 | 0.5729 |

| 1.44 | 0.6000 | 0.5789 |

| 1.45 | 0.6021 | 0.5849 |

| 1.46 | 0.6042 | 0.5910 |

| 1.47 | 0.6062 | 0.5970 |

| 1.48 | 0.6083 | 0.6031 |

| 1.49 | 0.6103 | 0.6092 |

| 1.50 | 0.6124 | 0.6153 |

| 1.51 | 0.6144 | 0.6215 |

| 1.52 | 0.6164 | 0.6276 |

| 1.53 | 0.6185 | 0.6338 |

| 1.54 | 0.6205 | 0.6400 |

| 1.55 | 0.6225 | 0.6463 |

| 1.56 | 0.6245 | 0.6525 |

| 1.57 | 0.6265 | 0.6588 |

| 1.58 | 0.6285 | 0.6650 |

| 1.59 | 0.6305 | 0.6714 |

| 1.60 | 0.6325 | 0.6777 |

| 1.61 | 0.6344 | 0.6840 |

| 1.62 | 0.6364 | 0.6904 |

| 1.63 | 0.6384 | 0.6968 |

| 1.64 | 0.6403 | 0.7032 |

| 1.65 | 0.6423 | 0.7096 |

| 1.66 | 0.6442 | 0.7160 |

| 1.67 | 0.6461 | 0.7225 |

| 1.68 | 0.6481 | 0.7290 |

| 1.69 | 0.6500 | 0.7355 |

| 1.70 | 0.6519 | 0.7420 |

| 1.71 | 0.6538 | 0.7485 |

| 1.72 | 0.6557 | 0.7551 |

| 1.73 | 0.6576 | 0.7617 |

| 1.74 | 0.6595 | 0.7683 |

| 1.75 | 0.6614 | 0.7749 |

| 1.76 | 0.6633 | 0.7815 |

| 1.77 | 0.6652 | 0.7882 |

| 1.78 | 0.6671 | 0.7948 |

| 1.79 | 0.6690 | 0.8015 |

| 1.80 | 0.6708 | 0.8082 |

| 1.81 | 0.6727 | 0.8150 |

| 1.82 | 0.6745 | 0.8217 |

| 1.83 | 0.6764 | 0.8285 |

| 1.84 | 0.6782 | 0.8353 |

| 1.85 | 0.6801 | 0.8421 |

| 1.86 | 0.6819 | 0.8489 |

| 1.87 | 0.6837 | 0.8557 |

| 1.88 | 0.6856 | 0.8626 |

| 1.89 | 0.6874 | 0.8694 |

| 1.90 | 0.6892 | 0.8763 |

| 1.91 | 0.6910 | 0.8832 |

| 1.92 | 0.6928 | 0.8902 |

| 1.93 | 0.6946 | 0.8971 |

| 1.94 | 0.6964 | 0.9041 |

| 1.95 | 0.6982 | 0.9111 |

| 1.96 | 0.7000 | 0.9181 |

| 1.97 | 0.7018 | 0.9251 |

| 1.98 | 0.7036 | 0.9321 |

| 1.99 | 0.7053 | 0.9392 |

| 2.00 | 0.7071 | 0.9462 |

| 2.01 | 0.7089 | 0.9533 |

| 2.02 | 0.7106 | 0.9604 |

| 2.03 | 0.7124 | 0.9676 |

| 2.04 | 0.7141 | 0.9747 |

| 2.05 | 0.7159 | 0.9819 |

| 2.06 | 0.7176 | 0.9890 |

| 2.07 | 0.7194 | 0.9962 |

| 2.08 | 0.7211 | 1.0034 |

| 2.09 | 0.7228 | 1.0107 |

| 2.10 | 0.7246 | 1.0179 |

| 2.11 | 0.7263 | 1.0252 |

| 2.12 | 0.7280 | 1.0325 |

| 2.13 | 0.7297 | 1.0398 |

| 2.14 | 0.7314 | 1.0471 |

| 2.15 | 0.7331 | 1.0544 |

| 2.16 | 0.7348 | 1.0618 |

| 2.17 | 0.7365 | 1.0691 |

| 2.18 | 0.7382 | 1.0765 |

| 2.19 | 0.7399 | 1.0839 |

| 2.20 | 0.7416 | 1.0913 |

| 2.21 | 0.7433 | 1.0987 |

| 2.22 | 0.7450 | 1.1062 |

| 2.23 | 0.7467 | 1.1137 |

| 2.24 | 0.7483 | 1.1211 |

| 2.25 | 0.7500 | 1.1286 |

| 2.26 | 0.7517 | 1.1362 |

| 2.27 | 0.7533 | 1.1437 |

| 2.28 | 0.7550 | 1.1512 |

| 2.29 | 0.7566 | 1.1588 |

| 2.30 | 0.7583 | 1.1664 |

| 2.31 | 0.7599 | 1.1740 |

| 2.32 | 0.7616 | 1.1816 |

| 2.33 | 0.7632 | 1.1892 |

| 2.34 | 0.7649 | 1.1969 |

| 2.35 | 0.7665 | 1.2046 |

| 2.36 | 0.7681 | 1.2122 |

| 2.37 | 0.7697 | 1.2199 |

| 2.38 | 0.7714 | 1.2276 |

| 2.39 | 0.7730 | 1.2354 |

| 2.40 | 0.7746 | 1.2431 |

| 2.41 | 0.7762 | 1.2509 |

| 2.42 | 0.7778 | 1.2587 |

| 2.43 | 0.7794 | 1.2665 |

| 2.44 | 0.7810 | 1.2743 |

| 2.45 | 0.7826 | 1.2821 |

| 2.46 | 0.7842 | 1.2899 |

| 2.47 | 0.7858 | 1.2978 |

| 2.48 | 0.7874 | 1.3057 |

| 2.49 | 0.7890 | 1.3136 |

| 2.50 | 0.7906 | 1.3215 |

| 2.51 | 0.7921 | 1.3294 |

| 2.52 | 0.7937 | 1.3373 |

| 2.53 | 0.7953 | 1.3453 |

| 2.54 | 0.7969 | 1.3532 |

| 2.55 | 0.7984 | 1.3612 |

| 2.56 | 0.8000 | 1.3692 |

| 2.57 | 0.8016 | 1.3772 |

| 2.58 | 0.8031 | 1.3853 |

| 2.59 | 0.8047 | 1.3933 |

| 2.60 | 0.8062 | 1.4014 |

| 2.61 | 0.8078 | 1.4095 |

| 2.62 | 0.8093 | 1.4176 |

| 2.63 | 0.8109 | 1.4257 |

| 2.64 | 0.8124 | 1.4338 |

| 2.65 | 0.8139 | 1.4419 |

| 2.66 | 0.8155 | 1.4501 |

| 2.67 | 0.8170 | 1.4583 |

| 2.68 | 0.8185 | 1.4664 |

| 2.69 | 0.8201 | 1.4746 |

| 2.70 | 0.8216 | 1.4829 |

| 2.71 | 0.8231 | 1.4911 |

| 2.72 | 0.8246 | 1.4993 |

| 2.73 | 0.8261 | 1.5076 |

| 2.74 | 0.8276 | 1.5159 |

| 2.75 | 0.8292 | 1.5242 |

| 2.76 | 0.8307 | 1.5325 |

| 2.77 | 0.8322 | 1.5408 |

| 2.78 | 0.8337 | 1.5491 |

| 2.79 | 0.8352 | 1.5575 |

| 2.80 | 0.8367 | 1.5658 |

| 2.81 | 0.8382 | 1.5742 |

| 2.82 | 0.8396 | 1.5826 |

| 2.83 | 0.8411 | 1.5910 |

| 2.84 | 0.8426 | 1.5995 |

| 2.85 | 0.8441 | 1.6079 |

| 2.86 | 0.8456 | 1.6164 |

| 2.87 | 0.8471 | 1.6248 |

| 2.88 | 0.8485 | 1.6333 |

| 2.89 | 0.8500 | 1.6418 |

| 2.90 | 0.8515 | 1.6503 |

| 2.91 | 0.8529 | 1.6589 |

| 2.92 | 0.8544 | 1.6674 |

| 2.93 | 0.8559 | 1.6760 |

| 2.94 | 0.8573 | 1.6845 |

| 2.95 | 0.8588 | 1.6931 |

| 2.96 | 0.8602 | 1.7017 |

| 2.97 | 0.8617 | 1.7103 |

| 2.98 | 0.8631 | 1.7190 |

| 2.99 | 0.8646 | 1.7276 |

| 3.00 | 0.8660 | 1.7363 |

| 3.01 | 0.8675 | 1.7450 |

| 3.02 | 0.8689 | 1.7536 |

| 3.03 | 0.8703 | 1.7623 |

| 3.04 | 0.8718 | 1.7711 |

| 3.05 | 0.8732 | 1.7798 |

| 3.06 | 0.8746 | 1.7885 |

| 3.07 | 0.8761 | 1.7973 |

| 3.08 | 0.8775 | 1.8061 |

| 3.09 | 0.8789 | 1.8149 |

| 3.10 | 0.8803 | 1.8237 |

| 3.11 | 0.8818 | 1.8325 |

| 3.12 | 0.8832 | 1.8413 |

| 3.13 | 0.8846 | 1.8502 |

| 3.14 | 0.8860 | 1.8590 |

| 3.15 | 0.8874 | 1.8679 |

| 3.16 | 0.8888 | 1.8768 |

| 3.17 | 0.8902 | 1.8857 |

| 3.18 | 0.8916 | 1.8946 |

| 3.19 | 0.8930 | 1.9035 |

| 3.20 | 0.8944 | 1.9125 |

| 3.21 | 0.8958 | 1.9214 |

| 3.22 | 0.8972 | 1.9304 |

| 3.23 | 0.8986 | 1.9394 |

| 3.24 | 0.9000 | 1.9484 |

| 3.25 | 0.9014 | 1.9574 |

| 3.26 | 0.9028 | 1.9664 |

| 3.27 | 0.9042 | 1.9755 |

| 3.28 | 0.9055 | 1.9845 |

| 3.29 | 0.9069 | 1.9936 |

| 3.30 | 0.9083 | 2.0027 |

| 3.31 | 0.9097 | 2.0118 |

| 3.32 | 0.9110 | 2.0209 |

| 3.33 | 0.9124 | 2.0300 |

| 3.34 | 0.9138 | 2.0392 |

| 3.35 | 0.9152 | 2.0483 |

| 3.36 | 0.9165 | 2.0575 |

| 3.37 | 0.9179 | 2.0667 |

| 3.38 | 0.9192 | 2.0758 |

| 3.39 | 0.9206 | 2.0851 |

| 3.40 | 0.9220 | 2.0943 |

| 3.41 | 0.9233 | 2.1035 |

| 3.42 | 0.9247 | 2.1128 |

| 3.43 | 0.9260 | 2.1220 |

| 3.44 | 0.9274 | 2.1313 |

| 3.45 | 0.9287 | 2.1406 |

| 3.46 | 0.9301 | 2.1499 |

| 3.47 | 0.9314 | 2.1592 |

| 3.48 | 0.9327 | 2.1685 |

| 3.49 | 0.9341 | 2.1779 |

| 3.50 | 0.9354 | 2.1872 |

| 3.51 | 0.9367 | 2.1966 |

| 3.52 | 0.9381 | 2.2060 |

| 3.53 | 0.9394 | 2.2154 |

| 3.54 | 0.9407 | 2.2248 |

| 3.55 | 0.9421 | 2.2342 |

| 3.56 | 0.9434 | 2.2436 |

| 3.57 | 0.9447 | 2.2531 |

| 3.58 | 0.9460 | 2.2625 |

| 3.59 | 0.9474 | 2.2720 |

| 3.60 | 0.9487 | 2.2815 |

| 3.61 | 0.9500 | 2.2910 |

| 3.62 | 0.9513 | 2.3005 |

| 3.63 | 0.9526 | 2.3100 |

| 3.64 | 0.9539 | 2.3196 |

| 3.65 | 0.9552 | 2.3291 |

| 3.66 | 0.9566 | 2.3387 |

| 3.67 | 0.9579 | 2.3483 |

| 3.68 | 0.9592 | 2.3578 |

| 3.69 | 0.9605 | 2.3675 |

| 3.70 | 0.9618 | 2.3771 |

| 3.71 | 0.9631 | 2.3867 |

| 3.72 | 0.9644 | 2.3963 |

| 3.73 | 0.9657 | 2.4060 |

| 3.74 | 0.9670 | 2.4157 |

| 3.75 | 0.9682 | 2.4254 |

| 3.76 | 0.9695 | 2.4350 |

| 3.77 | 0.9708 | 2.4448 |

| 3.78 | 0.9721 | 2.4545 |

| 3.79 | 0.9734 | 2.4642 |

| 3.80 | 0.9747 | 2.4740 |

| 3.81 | 0.9760 | 2.4837 |

| 3.82 | 0.9772 | 2.4935 |

| 3.83 | 0.9785 | 2.5033 |

| 3.84 | 0.9798 | 2.5131 |

| 3.85 | 0.9811 | 2.5229 |

| 3.86 | 0.9823 | 2.5327 |

| 3.87 | 0.9836 | 2.5425 |

| 3.88 | 0.9849 | 2.5524 |

| 3.89 | 0.9862 | 2.5623 |

| 3.90 | 0.9874 | 2.5721 |

| 3.91 | 0.9887 | 2.5820 |

| 3.92 | 0.9899 | 2.5919 |

| 3.93 | 0.9912 | 2.6018 |

| 3.94 | 0.9925 | 2.6118 |

| 3.95 | 0.9937 | 2.6217 |

| 3.96 | 0.9950 | 2.6316 |

| 3.97 | 0.9962 | 2.6416 |

| 3.98 | 0.9975 | 2.6516 |

| 3.99 | 0.9987 | 2.6616 |

| 4.00 | 1.0000 | 2.6716 |

| 4.01 | 1.0012 | 2.6816 |

| 4.02 | 1.0025 | 2.6916 |

| 4.03 | 1.0037 | 2.7016 |

| 4.04 | 1.0050 | 2.7117 |

| 4.05 | 1.0062 | 2.7218 |

| 4.06 | 1.0075 | 2.7318 |

| 4.07 | 1.0087 | 2.7419 |

| 4.08 | 1.0100 | 2.7520 |

| 4.09 | 1.0112 | 2.7621 |

| 4.10 | 1.0124 | 2.7722 |

| 4.11 | 1.0137 | 2.7824 |

| 4.12 | 1.0149 | 2.7925 |

| 4.13 | 1.0161 | 2.8027 |

| 4.14 | 1.0173 | 2.8129 |

| 4.15 | 1.0186 | 2.8231 |

| 4.16 | 1.0198 | 2.8333 |

| 4.17 | 1.0210 | 2.8435 |

| 4.18 | 1.0223 | 2.8537 |

| 4.19 | 1.0235 | 2.8639 |

| 4.20 | 1.0247 | 2.8742 |

| 4.21 | 1.0259 | 2.8844 |

| 4.22 | 1.0271 | 2.8947 |

| 4.23 | 1.0283 | 2.9050 |

| 4.24 | 1.0296 | 2.9153 |

| 4.25 | 1.0308 | 2.9256 |

| 4.26 | 1.0320 | 2.9359 |

| 4.27 | 1.0332 | 2.9462 |

| 4.28 | 1.0344 | 2.9566 |

| 4.29 | 1.0356 | 2.9669 |

| 4.30 | 1.0368 | 2.9773 |

| 4.31 | 1.0380 | 2.9877 |

| 4.32 | 1.0392 | 2.9981 |

| 4.33 | 1.0404 | 3.0085 |

| 4.34 | 1.0416 | 3.0189 |

| 4.35 | 1.0428 | 3.0293 |

| 4.36 | 1.0440 | 3.0398 |

| 4.37 | 1.0452 | 3.0502 |

| 4.38 | 1.0464 | 3.0607 |

| 4.39 | 1.0476 | 3.0712 |

| 4.40 | 1.0488 | 3.0816 |

| 4.41 | 1.0500 | 3.0921 |

| 4.42 | 1.0512 | 3.1027 |

| 4.43 | 1.0524 | 3.1132 |

| 4.44 | 1.0536 | 3.1237 |

| 4.45 | 1.0548 | 3.1343 |

| 4.46 | 1.0559 | 3.1448 |

| 4.47 | 1.0571 | 3.1554 |

| 4.48 | 1.0583 | 3.1660 |

| 4.49 | 1.0595 | 3.1766 |

| 4.50 | 1.0607 | 3.1872 |

| 4.51 | 1.0618 | 3.1978 |

| 4.52 | 1.0630 | 3.2084 |

| 4.53 | 1.0642 | 3.2191 |

| 4.54 | 1.0654 | 3.2297 |

| 4.55 | 1.0665 | 3.2404 |

| 4.56 | 1.0677 | 3.2511 |

| 4.57 | 1.0689 | 3.2618 |

| 4.58 | 1.0700 | 3.2725 |

| 4.59 | 1.0712 | 3.2832 |

| 4.60 | 1.0724 | 3.2939 |

| 4.61 | 1.0735 | 3.3046 |

| 4.62 | 1.0747 | 3.3154 |

| 4.63 | 1.0759 | 3.3261 |

| 4.64 | 1.0770 | 3.3369 |

| 4.65 | 1.0782 | 3.3477 |

| 4.66 | 1.0794 | 3.3585 |

| 4.67 | 1.0805 | 3.3693 |

| 4.68 | 1.0817 | 3.3801 |

| 4.69 | 1.0828 | 3.3909 |

| 4.70 | 1.0840 | 3.4018 |

| 4.71 | 1.0851 | 3.4126 |

| 4.72 | 1.0863 | 3.4235 |

| 4.73 | 1.0874 | 3.4344 |

| 4.74 | 1.0886 | 3.4452 |

| 4.75 | 1.0897 | 3.4561 |

| 4.76 | 1.0909 | 3.4670 |

| 4.77 | 1.0920 | 3.4780 |

| 4.78 | 1.0932 | 3.4889 |

| 4.79 | 1.0943 | 3.4998 |

| 4.80 | 1.0954 | 3.5108 |

| 4.81 | 1.0966 | 3.5218 |

| 4.82 | 1.0977 | 3.5327 |

| 4.83 | 1.0989 | 3.5437 |

| 4.84 | 1.1000 | 3.5547 |

| 4.85 | 1.1011 | 3.5657 |

| 4.86 | 1.1023 | 3.5768 |

| 4.87 | 1.1034 | 3.5878 |

| 4.88 | 1.1045 | 3.5988 |

| 4.89 | 1.1057 | 3.6099 |

| 4.90 | 1.1068 | 3.6210 |

| 4.91 | 1.1079 | 3.6320 |

| 4.92 | 1.1091 | 3.6431 |

| 4.93 | 1.1102 | 3.6542 |

| 4.94 | 1.1113 | 3.6654 |

| 4.95 | 1.1124 | 3.6765 |

| 4.96 | 1.1136 | 3.6876 |

| 4.97 | 1.1147 | 3.6988 |

| 4.98 | 1.1158 | 3.7099 |

| 4.99 | 1.1169 | 3.7211 |

| 5.00 | 1.1180 | 3.7323 |

Метод Рунге-Кутта 4-го порядка:

Алгоритм решения дифференциального уравнения

методом Рунге-Кутта 4-го порядка.

Файл-функция:

f=faaan(x)=x;

Файл-программа:

a=5;=10;=0.05;=0;=5;=0;=0;('Таблица значений по МРГ\n');('--------------------------\n');('| t | x | y |\n');

fprintf('--------------------------\n');=t0;=(a/b)*sqrt(t);('| %4.2f | %4.4f | %4.4f |\n',t,x,y);t<tk-h

f=faaan(x);

k=h*f;

s=k;

y=y+k/2;

t=t+h/2;

x=(a/b)*sqrt(t);

for j=1:2

f=faaan(x);

k=h*f;

s=s+2*k;

y=y1+k/2;

end

t=t+h/2;

x=(a/b)*sqrt(t);

y=y+k/2;

f=faaan(x);

k=h*f;

s=s+k;

y=y1+s/6;

fprintf('| %4.2f | %4.4f | %4.4f |\n',t,x,y);

plot(t,x,'r*')

grid on

hold on

plot(t,y,'k+')

xlabel('t');

ylabel('y');

y1=y;

end

Результат работы программы:

Таблица значений по МРГ

------------------------------

| t | x | y |

-----------------------------

| 0.00 | 0.0000 | 0.0000 |

| 0.05 | 0.1118 | 0.0036 |

| 0.10 | 0.1581 | 0.0104 |

| 0.15 | 0.1936 | 0.0192 |

| 0.20 | 0.2236 | 0.0297 |

| 0.25 | 0.2500 | 0.0415 |

| 0.30 | 0.2739 | 0.0546 |

| 0.35 | 0.2958 | 0.0689 |

| 0.40 | 0.3162 | 0.0842 |

| 0.45 | 0.3354 | 0.1005 |

| 0.50 | 0.3536 | 0.1177 |

| 0.55 | 0.3708 | 0.1358 |

| 0.60 | 0.3873 | 0.1548 |

| 0.65 | 0.4031 | 0.1745 |

| 0.70 | 0.4183 | 0.1951 |

| 0.75 | 0.4330 | 0.2163 |

| 0.80 | 0.4472 | 0.2384 |

| 0.85 | 0.4610 | 0.2611 |

| 0.90 | 0.4743 | 0.2844 |

| 0.95 | 0.4873 | 0.3085 |

| 1.00 | 0.5000 | 0.3332 |

| 1.05 | 0.5123 | 0.3585 |

| 1.10 | 0.5244 | 0.3844 |

| 1.15 | 0.5362 | 0.4109 |

| 1.20 | 0.5477 | 0.4380 |

| 1.25 | 0.5590 | 0.4657 |

| 1.30 | 0.5701 | 0.4939 |

| 1.35 | 0.5809 | 0.5227 |

| 1.40 | 0.5916 | 0.5520 |

| 1.45 | 0.6021 | 0.5818 |

| 1.50 | 0.6124 | 0.6122 |

| 1.55 | 0.6225 | 0.6431 |

| 1.60 | 0.6325 | 0.6745 |

| 1.65 | 0.6423 | 0.7063 |

| 1.70 | 0.6519 | 0.7387 |

| 1.75 | 0.6614 | 0.7715 |

| 1.80 | 0.6708 | 0.8048 |

| 1.85 | 0.6801 | 0.8386 |

| 1.90 | 0.6892 | 0.8728 |

| 1.95 | 0.6982 | 0.9075 |

| 2.00 | 0.7071 | 0.9426 |

| 2.05 | 0.7159 | 0.9782 |

| 2.10 | 0.7246 | 1.0142 |

| 2.15 | 0.7331 | 1.0507 |

| 2.20 | 0.7416 | 1.0875 |

| 2.25 | 0.7500 | 1.1248 |

| 2.30 | 0.7583 | 1.1625 |

| 2.35 | 0.7665 | 1.2007 |

| 2.40 | 0.7746 | 1.2392 |

| 2.45 | 0.7826 | 1.2781 |

| 2.50 | 0.7906 | 1.3175 |

| 2.55 | 0.7984 | 1.3572 |

| 2.60 | 0.8062 | 1.3973 |

| 2.65 | 0.8139 | 1.4378 |

| 2.70 | 0.8216 | 1.4787 |

| 2.75 | 0.8292 | 1.5200 |

| 2.80 | 0.8367 | 1.5616 |

| 2.85 | 0.8441 | 1.6036 |

| 2.90 | 0.8515 | 1.6460 |

| 2.95 | 0.8588 | 1.6888 |

| 3.00 | 0.8660 | 1.7319 |

| 3.05 | 0.8732 | 1.7754 |

| 3.10 | 0.8803 | 1.8192 |

| 3.15 | 0.8874 | 1.8634 |

| 3.20 | 0.8944 | 1.9080 |

| 3.25 | 0.9014 | 1.9528 |

| 3.30 | 0.9083 | 1.9981 |

| 3.35 | 0.9152 | 2.0437 |

| 3.40 | 0.9220 | 2.0896 |

| 3.45 | 0.9287 | 2.1359 |

| 3.50 | 0.9354 | 2.1825 |

| 3.55 | 0.9421 | 2.2294 |

| 3.60 | 0.9487 | 2.2767 |

| 3.65 | 0.9552 | 2.3243 |

| 3.70 | 0.9618 | 2.3722 |

| 3.75 | 0.9682 | 2.4205 |

| 3.80 | 0.9747 | 2.4690 |

| 3.85 | 0.9811 | 2.5179 |

| 3.90 | 0.9874 | 2.5671 |

| 3.95 | 0.9937 | 2.6167 |

| 4.00 | 1.0000 | 2.6665 |

| 4.05 | 1.0062 | 2.7167 |

| 4.10 | 1.0124 | 2.7671 |

| 4.15 | 1.0186 | 2.8179 |

| 4.20 | 1.0247 | 2.8690 |

| 4.25 | 1.0308 | 2.9204 |

| 4.30 | 1.0368 | 2.9721 |

| 4.35 | 1.0428 | 3.0241 |

| 4.40 | 1.0488 | 3.0763 |

| 4.45 | 1.0548 | 3.1289 |

| 4.50 | 1.0607 | 3.1818 |

| 4.55 | 1.0665 | 3.2350 |

| 4.60 | 1.0724 | 3.2885 |

| 4.65 | 1.0782 | 3.3422 |

| 4.70 | 1.0840 | 3.3963 |

| 4.75 | 1.0897 | 3.4506 |

| 4.80 | 1.0954 | 3.5053 |

| 4.85 | 1.1011 | 3.5602 |

| 4.90 | 1.1068 | 3.6154 |

| 4.95 | 1.1124 | 3.6709 |

б) с использованием солвера ode45;

Файл-функция:

function f=fqn(t,y)

b=2;=5;=(a/b)*sqrt(t);=x;

Файл-программа:

[T,Y]=ode45('fqn',[0:0.05:5],[0]);

[T,Y](T,Y,'b*')on('t');('y');

Результат работы программы:

ans =

0 0

0.0500 0.0183

0.1000 0.0524

0.1500 0.0965

0.2000 0.1488

0.2500 0.2080

0.3000 0.2735

0.3500 0.3448

0.4000 0.4213

0.4500 0.5028

0.5000 0.5889

0.5500 0.6795

0.6000 0.7743

0.6500 0.8731

0.7000 0.9758

0.7500 1.0822

0.8000 1.1922

0.8500 1.3058

0.9000 1.4227

0.9500 1.5429

1.0000 1.6664

1.0500 1.7929

1.1000 1.9225

1.1500 2.0551

1.2000 2.1906

1.2500 2.3289

1.3000 2.4701

1.3500 2.6139

1.4000 2.7605

1.4500 2.9097

1.5000 3.0615

1.5500 3.2159

1.6000 3.3728

1.6500 3.5321

1.7000 3.6939

1.7500 3.8581

1.8000 4.0246

1.8500 4.1935

1.9000 4.3646

1.9500 4.5381

2.0000 4.7137

2.0500 4.8916

2.1000 5.0717

2.1500 5.2539

2.2000 5.4382

2.2500 5.6247

2.3000 5.8132

2.3500 6.0038

2.4000 6.1965

2.4500 6.3911

2.5000 6.5878

2.5500 6.7864

2.6000 6.9870

2.6500 7.1895

2.7000 7.3939

2.7500 7.6003

2.8000 7.8085

2.8500 8.0186

2.9000 8.2305

2.9500 8.4443

3.0000 8.6599

3.0500 8.8773

3.1000 9.0965

3.1500 9.3175

3.2000 9.5402

3.2500 9.7647

3.3000 9.9909

3.3500 10.2189

3.4000 10.4485

3.4500 10.6798

3.5000 10.9128

3.5500 11.1475

3.6000 11.3839

3.6500 11.6219

3.7000 11.8615

3.7500 12.1027

3.8000 12.3456

3.8500 12.5901

3.9000 12.8361

3.9500 13.0838

4.0000 13.3330

4.0500 13.5838

4.1000 13.8361

4.1500 14.0900

4.2000 14.3454

4.2500 14.6023

4.3000 14.8608

4.3500 15.1208

4.4000 15.3822

4.4500 15.6452

4.5000 15.9096

4.5500 16.1755

4.6000 16.4428

4.6500 16.7117

4.7000 16.9819

4.7500 17.2537

4.8000 17.5268

4.8500 17.8014

4.9000 18.0774

4.9500 18.3548

5.0000 18.6336

в) с использованием системной функции feval

Файл-функция:

function f=fyn(x)=10;=5;=0:0.05:5;=(a/b).*sqrt(t);=x;

Файл-программа:

figure('Таблица значений (feval)\n');('......................\n');('| t | x | y |\n');('......................\n');=5;=10;=0:0.05:5;=(a/b).*sqrt(t);=feval('fyn');=[t;x;y];('| %4.2f | %4.2f | %4.2f |\n',v);(t,x,'b*')onon(t,y,'r+')

xlabel('x');('y');

Результат работы программы:

Таблица значений (feval)

......................

| t | x | y |

......................

| 0.00 | 0.00 | 0.00 |

| 0.05 | 0.11 | 0.11 |

| 0.10 | 0.16 | 0.16 |

| 0.15 | 0.19 | 0.19 |

| 0.20 | 0.22 | 0.22 |

| 0.25 | 0.25 | 0.25 |

| 0.30 | 0.27 | 0.27 |

| 0.35 | 0.30 | 0.30 |

| 0.40 | 0.32 | 0.32 |

| 0.45 | 0.34 | 0.34 |

| 0.50 | 0.35 | 0.35 |

| 0.55 | 0.37 | 0.37 |

| 0.60 | 0.39 | 0.39 |

| 0.65 | 0.40 | 0.40 |

| 0.70 | 0.42 | 0.42 |

| 0.75 | 0.43 | 0.43 |

| 0.80 | 0.45 | 0.45 |

| 0.85 | 0.46 | 0.46 |

| 0.90 | 0.47 | 0.47 |

| 0.95 | 0.49 | 0.49 |

| 1.00 | 0.50 | 0.50 |

| 1.05 | 0.51 | 0.51 |

| 1.10 | 0.52 | 0.52 |

| 1.15 | 0.54 | 0.54 |

| 1.20 | 0.55 | 0.55 |

| 1.25 | 0.56 | 0.56 |

| 1.30 | 0.57 | 0.57 |

| 1.35 | 0.58 | 0.58 |

| 1.40 | 0.59 | 0.59 |

| 1.45 | 0.60 | 0.60 |

| 1.50 | 0.61 | 0.61 |

| 1.55 | 0.62 | 0.62 |

| 1.60 | 0.63 | 0.63 |

| 1.65 | 0.64 | 0.64 |

| 1.70 | 0.65 | 0.65 |

| 1.75 | 0.66 | 0.66 |

| 1.80 | 0.67 | 0.67 |

| 1.85 | 0.68 | 0.68 |

| 1.90 | 0.69 | 0.69 |

| 1.95 | 0.70 | 0.70 |

| 2.00 | 0.71 | 0.71 |

| 2.05 | 0.72 | 0.72 |

| 2.10 | 0.72 | 0.72 |

| 2.15 | 0.73 | 0.73 |

| 2.20 | 0.74 | 0.74 |

| 2.25 | 0.75 | 0.75 |

| 2.30 | 0.76 | 0.76 |

| 2.35 | 0.77 | 0.77 |

| 2.40 | 0.77 | 0.77 |

| 2.45 | 0.78 | 0.78 |

| 2.50 | 0.79 | 0.79 |

| 2.55 | 0.80 | 0.80 |

| 2.60 | 0.81 | 0.81 |

| 2.65 | 0.81 | 0.81 |

| 2.70 | 0.82 | 0.82 |

| 2.75 | 0.83 | 0.83 |

| 2.80 | 0.84 | 0.84 |

| 2.85 | 0.84 | 0.84 |

| 2.90 | 0.85 | 0.85 |

| 2.95 | 0.86 | 0.86 |

| 3.00 | 0.87 | 0.87 |

| 3.05 | 0.87 | 0.87 |

| 3.10 | 0.88 | 0.88 |

| 3.15 | 0.89 | 0.89 |

| 3.20 | 0.89 | 0.89 |

| 3.25 | 0.90 | 0.90 |

| 3.30 | 0.91 | 0.91 |

| 3.35 | 0.92 | 0.92 |

| 3.40 | 0.92 | 0.92 |

| 3.45 | 0.93 | 0.93 |

| 3.50 | 0.94 | 0.94 |

| 3.55 | 0.94 | 0.94 |

| 3.60 | 0.95 | 0.95 |

| 3.65 | 0.96 | 0.96 |

| 3.70 | 0.96 | 0.96 |

| 3.75 | 0.97 | 0.97 |

| 3.80 | 0.97 | 0.97 |

| 3.85 | 0.98 | 0.98 |

| 3.90 | 0.99 | 0.99 |

| 3.95 | 0.99 | 0.99 |

| 4.00 | 1.00 | 1.00 |

| 4.05 | 1.01 | 1.01 |

| 4.10 | 1.01 | 1.01 |

| 4.15 | 1.02 | 1.02 |

| 4.20 | 1.02 | 1.02 |

| 4.25 | 1.03 | 1.03 |

| 4.30 | 1.04 | 1.04 |

| 4.35 | 1.04 | 1.04 |

| 4.40 | 1.05 | 1.05 |

| 4.45 | 1.05 | 1.05 |

| 4.50 | 1.06 | 1.06 |

| 4.55 | 1.07 | 1.07 |

| 4.60 | 1.07 | 1.07 |

| 4.65 | 1.08 | 1.08 |

| 4.70 | 1.08 | 1.08 |

| 4.75 | 1.09 | 1.09 |

| 4.80 | 1.10 | 1.10 |

| 4.85 | 1.10 | 1.10 |

| 4.90 | 1.11 | 1.11 |

| 4.95 | 1.11 | 1.11 |

| 5.00 | 1.12 | 1.12 |

>>

МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ Федеральное государственное автономное образовательное учреждение высшего профессионального образова

Больше работ по теме: