{"id":1102,"date":"2021-11-15T20:57:11","date_gmt":"2021-11-16T04:57:11","guid":{"rendered":"https:\/\/gantovnik.com\/bio-tips\/?p=1102"},"modified":"2021-11-16T15:06:09","modified_gmt":"2021-11-16T23:06:09","slug":"192-4th-order-runge-kutta-method","status":"publish","type":"post","link":"https:\/\/gantovnik.com\/bio-tips\/2021\/11\/192-4th-order-runge-kutta-method\/","title":{"rendered":"#192 4th order Runge-Kutta method"},"content":{"rendered":"[et_pb_section admin_label=”section”]\n\t\t\t[et_pb_row admin_label=”row”]\n\t\t\t\t[et_pb_column type=”4_4″][et_pb_text admin_label=”Text”]\n
\r\n# 4th order Runge-Kutta\r\nimport os\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\nimport math\r\nos.chdir(r'D:\\projects\\wordpress\\ex192')\r\nos.getcwd()\r\n# initialization\r\na=0.0\r\nb=10.0\r\nn=10000\r\nydumb=np.zeros((2),float)\r\ny=np.zeros((2),float)\r\nfReturn=np.zeros((2),float)\r\nk1=np.zeros((2),float)\r\nk2=np.zeros((2),float)\r\nk3=np.zeros((2),float)\r\nk4=np.zeros((2),float)\r\ny[0]=3.0\r\ny[1]=-5.0\r\nt=a\r\nh=(b-a)\/n\r\n\r\ndef f(t,y):\r\n    # force function\r\n    fReturn[0] = y[1]\r\n    fReturn[1] = -100.0 *y[0] -2.0*y[1] + 10.0*math.sin(3*t)\r\n    return fReturn\r\n\r\ndef rk4(t,h,n):\r\n    k1=[0]*(n)\r\n    k2=[0]*(n)\r\n    k3=[0]*(n)\r\n    k4=[0]*(n)\r\n    fR=[0]*(n)\r\n    ydumb=[0]*(n)\r\n    # return RHS\r\n    fR=f(t,y)\r\n    for i in range(0,n):\r\n        k1[i]=h*fR[i]\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k1[i]\/2\r\n    k2=h*f(t+h\/2,ydumb)\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k2[i]\/2\r\n    k3=h*f(t+h\/2,ydumb)\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k3[i]\/2\r\n    k4=h*f(t+h,ydumb)\r\n    for i in range(0,2):\r\n        y[i]=y[i] + (k1[i] + 2.0* (k2[i] + k3[i]) + k4[i])\/6.0\r\n    return y\r\n\r\ntarray=[]\r\ny1array=[]\r\ny2array=[]\r\nwhile (t < b):\r\n    if ((t+h) > b):\r\n        h = b-t\r\n    y=rk4(t,h,2)\r\n    t = t+h\r\n    tarray.append(t)\r\n    y1array.append(y[0])\r\n    y2array.append(y[1])\r\n\r\nplt.plot(tarray,y1array,color='r',label="y1")\r\nplt.plot(tarray,y2array,color='g',label="y2")\r\nplt.xlim(a, b)\r\nplt.legend(["4nd Order RK"], loc=1)\r\nplt.xlabel('t', fontsize=17)\r\nplt.ylabel('y(t)', fontsize=17)\r\nplt.legend()\r\nplt.tight_layout()\r\nplt.savefig("example192_1.png", dpi=300)\r\nplt.show()\r\nplt.close()\r\nplt.plot(y1array,y2array)\r\nplt.legend(["4nd Order RK"], loc=1)\r\nplt.xlabel('y1', fontsize=17)\r\nplt.ylabel('y2', fontsize=17)\r\nplt.tight_layout()\r\nplt.savefig("example192_2.png", dpi=300)\r\nplt.show()\r\nplt.close()\r\n<\/pre>\n
\"\"<\/p>\n
\"\"[\/et_pb_text][\/et_pb_column]\n\t\t\t[\/et_pb_row]\n\t\t[\/et_pb_section]\n","protected":false},"excerpt":{"rendered":"
[code language=”python”] # 4th order Runge-Kutta import os import numpy as np import matplotlib.pyplot as plt import math os.chdir(r’D:\\projects\\wordpress\\ex192′) os.getcwd() # initialization a=0.0 b=10.0 n=10000 ydumb=np.zeros((2),float) y=np.zeros((2),float) fReturn=np.zeros((2),float) k1=np.zeros((2),float) k2=np.zeros((2),float) k3=np.zeros((2),float) k4=np.zeros((2),float) y[0]=3.0 y[1]=-5.0 t=a h=(b-a)\/n def f(t,y): # force function fReturn[0] = y[1] fReturn[1] = -100.0 *y[0] -2.0*y[1] + 10.0*math.sin(3*t) return fReturn def rk4(t,h,n): […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","_et_pb_use_builder":"on","_et_pb_old_content":"[code language=\"python\"]\r\n# 4th order Runge-Kutta\r\nimport os\r\nimport numpy as np\r\nimport matplotlib.pyplot as plt\r\nimport math\r\nos.chdir(r'D:projectswordpressex192')\r\nos.getcwd()\r\n# initialization\r\na=0.0\r\nb=10.0\r\nn=10000\r\nydumb=np.zeros((2),float)\r\ny=np.zeros((2),float)\r\nfReturn=np.zeros((2),float)\r\nk1=np.zeros((2),float)\r\nk2=np.zeros((2),float)\r\nk3=np.zeros((2),float)\r\nk4=np.zeros((2),float)\r\ny[0]=3.0\r\ny[1]=-5.0\r\nt=a\r\nh=(b-a)\/n\r\n\r\ndef f(t,y):\r\n    # force function\r\n    fReturn[0] = y[1]\r\n    fReturn[1] = -100.0 *y[0] -2.0*y[1] + 10.0*math.sin(3*t)\r\n    return fReturn\r\n\r\ndef rk4(t,h,n):\r\n    k1=[0]*(n)\r\n    k2=[0]*(n)\r\n    k3=[0]*(n)\r\n    k4=[0]*(n)\r\n    fR=[0]*(n)\r\n    ydumb=[0]*(n)\r\n    # return RHS\r\n    fR=f(t,y)\r\n    for i in range(0,n):\r\n        k1[i]=h*fR[i]\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k1[i]\/2\r\n    k2=h*f(t+h\/2,ydumb)\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k2[i]\/2\r\n    k3=h*f(t+h\/2,ydumb)\r\n    for i in range(0,n):\r\n        ydumb[i]=y[i] + k3[i]\/2\r\n    k4=h*f(t+h,ydumb)\r\n    for i in range(0,2):\r\n        y[i]=y[i] + (k1[i] + 2.0* (k2[i] + k3[i]) + k4[i])\/6.0\r\n    return y\r\n\r\ntarray=[]\r\ny1array=[]\r\ny2array=[]\r\nwhile (t < b):\r\n    if ((t+h) > b):\r\n        h = b-t\r\n    y=rk4(t,h,2)\r\n    t = t+h\r\n    tarray.append(t)\r\n    y1array.append(y[0])\r\n    y2array.append(y[1])\r\n\r\nplt.plot(tarray,y1array,color='r',label="y1")\r\nplt.plot(tarray,y2array,color='g',label="y2")\r\nplt.xlim(a, b)\r\nplt.legend(["4nd Order RK"], loc=1)\r\nplt.xlabel('t', fontsize=17)\r\nplt.ylabel('y(t)', fontsize=17)\r\nplt.legend()\r\nplt.tight_layout()\r\nplt.savefig("example192_1.png", dpi=300)\r\nplt.show()\r\nplt.close()\r\nplt.plot(y1array,y2array)\r\nplt.legend(["4nd Order RK"], loc=1)\r\nplt.xlabel('y1', fontsize=17)\r\nplt.ylabel('y2', fontsize=17)\r\nplt.tight_layout()\r\nplt.savefig("example192_2.png", dpi=300)\r\nplt.show()\r\nplt.close()\r\n[\/code]\r\n\r\n\"\"\r\n\r\n\"\"","_et_gb_content_width":"","_lmt_disableupdate":"yes","_lmt_disable":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-1102","post","type-post","status-publish","format-standard","hentry","category-python"],"modified_by":"gantovnik","jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p8bH0k-hM","jetpack_likes_enabled":true,"jetpack-related-posts":[{"id":210,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/2nd-order-runge-kutta-type-a\/","url_meta":{"origin":1102,"position":0},"title":"#44 2nd-order Runge-Kutta type A using python","author":"gantovnik","date":"2019-01-12","format":false,"excerpt":"[code language=\"python\"] import os import numpy as np import matplotlib.pyplot as plt os.chdir(r'D:\\projects\\wordpress\\ex44') os.getcwd() #2nd-order Runge-Kutta methods with A=1\/2 (type A) # dy\/dx=exp(-2x)-2y # y(0)=0.1, interval x=[0,2], step size = h=0.2 def feval(funcName, *args): return eval(funcName)(*args) def RK2A(func, yinit, x_range, h): m = len(yinit) n = int((x_range[-1] - x_range[0])\/h) x\u2026","rel":"","context":"In "numerical"","block_context":{"text":"numerical","link":"https:\/\/gantovnik.com\/bio-tips\/category\/numerical\/"},"img":{"alt_text":"example44","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example44-1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example44-1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example44-1.png?resize=525%2C300 1.5x"},"classes":[]},{"id":217,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/2nd-order-runge-kutta-type-b\/","url_meta":{"origin":1102,"position":1},"title":"#45  2nd-order Runge-Kutta type B using python","author":"gantovnik","date":"2019-01-12","format":false,"excerpt":"import os import numpy as np import matplotlib.pyplot as plt os.chdir(r'D:\\projects\\wordpress\\ex45') os.getcwd() #2nd-order Runge-Kutta methods with A=0 (type B) # dy\/dx=exp(-2x)-2y # y(0)=0.1, interval x=[0,2], step size = h=0.2 def feval(funcName, *args): return eval(funcName)(*args) def RK2B(func, yinit, x_range, h): m = len(yinit) n = int((x_range[-1] - x_range[0])\/h) x = x_range[0]\u2026","rel":"","context":"In "differential equations"","block_context":{"text":"differential equations","link":"https:\/\/gantovnik.com\/bio-tips\/category\/differential-equations\/"},"img":{"alt_text":"example45","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example45.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example45.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example45.png?resize=525%2C300 1.5x"},"classes":[]},{"id":220,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/2nd-order-runge-kutta-type-c\/","url_meta":{"origin":1102,"position":2},"title":"#46 2nd-order Runge-Kutta type C using python","author":"gantovnik","date":"2019-01-12","format":false,"excerpt":"import os import numpy as np import matplotlib.pyplot as plt os.chdir(r'D:\\projects\\wordpress\\ex46') os.getcwd() #2nd-order Runge-Kutta methods with A=1\/3 (type C) # dy\/dx=exp(-2x)-2y # y(0)=0.1, interval x=[0,2], step size = h=0.2 def feval(funcName, *args): return eval(funcName)(*args) def RK2C(func, yinit, x_range, h): m = len(yinit) n = int((x_range[-1] - x_range[0])\/h) x = x_range[0]\u2026","rel":"","context":"In "numerical"","block_context":{"text":"numerical","link":"https:\/\/gantovnik.com\/bio-tips\/category\/numerical\/"},"img":{"alt_text":"example46","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example46.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example46.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example46.png?resize=525%2C300 1.5x"},"classes":[]},{"id":190,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/coupled-damped-springs-jacobian-is-available\/","url_meta":{"origin":1102,"position":3},"title":"Coupled damped springs (Jacobian is available)","author":"gantovnik","date":"2019-01-10","format":false,"excerpt":"\u00a0 import os import numpy as np import matplotlib.pyplot as plt from scipy import integrate os.chdir(r'D:\\projects\\wordpress\\ex38') os.getcwd() def f(t, y, args): m1, k1, g1, m2, k2, g2 = args return [y[1], - k1\/m1 * y[0] + k2\/m1 * (y[2] - y[0]) - g1\/m1 * y[1], y[3], - k2\/m2 * (y[2]\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example38","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example38.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example38.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example38.png?resize=525%2C300 1.5x"},"classes":[]},{"id":187,"url":"https:\/\/gantovnik.com\/bio-tips\/2019\/01\/coupled-damped-springs\/","url_meta":{"origin":1102,"position":4},"title":"Coupled damped springs","author":"gantovnik","date":"2019-01-09","format":false,"excerpt":"\u00a0 import os import numpy as np import matplotlib.pyplot as plt from scipy import integrate os.chdir(r'D:\\projects\\wordpress\\ex37') os.getcwd() def f(t, y, args): m1, k1, g1, m2, k2, g2 = args return [y[1], - k1\/m1 * y[0] + k2\/m1 * (y[2] - y[0]) - g1\/m1 * y[1], y[3], - k2\/m2 * (y[2]\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"example37","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example37.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example37.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2019\/01\/example37.png?resize=525%2C300 1.5x"},"classes":[]},{"id":1261,"url":"https:\/\/gantovnik.com\/bio-tips\/2021\/12\/210-parametric-curve-in-3d-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2-2\/","url_meta":{"origin":1102,"position":5},"title":"#226 Finite difference method to solve ODE using python","author":"gantovnik","date":"2021-12-04","format":false,"excerpt":"Solve the following problem: d^2y\/dt^2=-4*y + 4*x Boundary conditions are y(0) = 0 and y'(\u03c0\/2) = 0. The exact solution of the problem is y = x \u2212 sin(2x*) y(0) = 0, y(i\u22121) \u2212 2*y(i) + y(i+1) \u2212 h^2*(\u22124*y(i) + 4*x(i)) = 0, i = 1,2,...,n \u2212 1 2*y(n\u22121) \u2212\u2026","rel":"","context":"In "python"","block_context":{"text":"python","link":"https:\/\/gantovnik.com\/bio-tips\/category\/python\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/12\/ex226.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/12\/ex226.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/12\/ex226.png?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/gantovnik.com\/bio-tips\/wp-content\/uploads\/2021\/12\/ex226.png?resize=700%2C400&ssl=1 2x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/1102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/comments?post=1102"}],"version-history":[{"count":0,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/posts\/1102\/revisions"}],"wp:attachment":[{"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/media?parent=1102"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/categories?post=1102"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gantovnik.com\/bio-tips\/wp-json\/wp\/v2\/tags?post=1102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}