{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-08-04T00:16:45.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-08-04T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":46084,"title":"Parabolic Partial Differential Equations: Explicit Method","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 294px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 147px; transform-origin: 407px 147px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI got this example from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003enumericalmethodsguy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  youtube video. He solves the problem step by step.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=p0V1eSlM2xo\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOn the second test: I varied the length of rod and the final time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efrom t = 0 and t = 102 seconds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pderod(L,tf)\r\n% defined constants\r\nk=54;                %W/(m*K)\r\np=7800;              %kg/m^3\r\nC=490;               %J/(kg*K)\r\nL0 = 0;               %meter \r\nnode = 5;            % segments\r\ndt = 3;              %second time steps\r\nt0=0;                %s initial time\r\nT0 = 100;           %C initial Temperature   \r\nTend = 25;          %C final Temperature\r\nTm1 =20;            %C\r\nTm = repelem(Tm1,node-1)'; %C middle Temperature \r\nTi=[T0;Tm;Tend];     % Celsius at Time 0second \r\nT0=Ti;               % Celsius at Time 0second \r\n\r\nend","test_suite":"%%\r\n% INPUTS\r\nL = .05;                     %meter\r\ntf = 9;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   53.9089   59.0725   65.9508   85.0000\r\n   20.0000   20.0000   34.3727   39.1307   70.0000\r\n   20.0000   20.0000   20.8983   27.2639   55.0000\r\n   20.0000   22.1193   22.4420   22.8719   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend\r\n%%\r\n% INPUTS\r\nL = .10;                     %meter\r\ntf = 102;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   28.4772   35.1579   40.5179   44.8926   48.5211   51.5762   54.1838   56.4375   58.4071   60.1460   61.6949   63.0856   64.3431   65.4874   66.5344   67.4973   68.3867   69.2115   69.9791   70.6957   71.3666   71.9963   72.5887   73.1472   73.6746   74.1735   74.6463   75.0947   75.5206   75.9256   76.3111   76.6782   77.0283   77.3622   85.0000\r\n   20.0000   20.0000   20.8983   22.3201   24.0280   25.8726   27.7607   29.6354   31.4629   33.2239   34.9089   36.5138   38.0384   39.4848   40.8559   42.1556   43.3879   44.5569   45.6666   46.7206   47.7226   48.6758   49.5832   50.4477   51.2717   52.0578   52.8081   53.5245   54.2090   54.8632   55.4887   56.0871   56.6596   57.2077   57.7323   70.0000\r\n   20.0000   20.0000   20.0561   20.2398   20.5707   21.0425   21.6372   22.3328   23.1075   23.9414   24.8173   25.7209   26.6402   27.5657   28.4898   29.4063   30.3106   31.1990   32.0688   32.9179   33.7447   34.5484   35.3281   36.0835   36.8145   37.5211   38.2035   38.8621   39.4971   40.1091   40.6987   41.2663   41.8126   42.3382   42.8437   55.0000\r\n   20.0000   20.5298   20.9474   21.2824   21.5658   21.8243   22.0780   22.3409   22.6218   22.9253   23.2528   23.6037   23.9760   24.3669   24.7729   25.1908   25.6173   26.0492   26.4837   26.9183   27.3508   27.7792   28.2020   28.6178   29.0256   29.4244   29.8135   30.1925   30.5609   30.9186   31.2653   31.6010   31.9257   32.2395   32.5425   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":227209,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-31T20:44:36.000Z","updated_at":"2026-05-30T15:28:07.000Z","published_at":"2020-07-31T21:36:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI got this example from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\\\"\u003e\u003cw:r\u003e\u003cw:t\u003enumericalmethodsguy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e  youtube video. He solves the problem step by step.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=p0V1eSlM2xo\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the second test: I varied the length of rod and the final time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efrom t = 0 and t = 102 seconds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"problems":[{"id":46084,"title":"Parabolic Partial Differential Equations: Explicit Method","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 294px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 147px; transform-origin: 407px 147px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI got this example from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003enumericalmethodsguy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  youtube video. He solves the problem step by step.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=p0V1eSlM2xo\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOn the second test: I varied the length of rod and the final time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efrom t = 0 and t = 102 seconds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pderod(L,tf)\r\n% defined constants\r\nk=54;                %W/(m*K)\r\np=7800;              %kg/m^3\r\nC=490;               %J/(kg*K)\r\nL0 = 0;               %meter \r\nnode = 5;            % segments\r\ndt = 3;              %second time steps\r\nt0=0;                %s initial time\r\nT0 = 100;           %C initial Temperature   \r\nTend = 25;          %C final Temperature\r\nTm1 =20;            %C\r\nTm = repelem(Tm1,node-1)'; %C middle Temperature \r\nTi=[T0;Tm;Tend];     % Celsius at Time 0second \r\nT0=Ti;               % Celsius at Time 0second \r\n\r\nend","test_suite":"%%\r\n% INPUTS\r\nL = .05;                     %meter\r\ntf = 9;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   53.9089   59.0725   65.9508   85.0000\r\n   20.0000   20.0000   34.3727   39.1307   70.0000\r\n   20.0000   20.0000   20.8983   27.2639   55.0000\r\n   20.0000   22.1193   22.4420   22.8719   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend\r\n%%\r\n% INPUTS\r\nL = .10;                     %meter\r\ntf = 102;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   28.4772   35.1579   40.5179   44.8926   48.5211   51.5762   54.1838   56.4375   58.4071   60.1460   61.6949   63.0856   64.3431   65.4874   66.5344   67.4973   68.3867   69.2115   69.9791   70.6957   71.3666   71.9963   72.5887   73.1472   73.6746   74.1735   74.6463   75.0947   75.5206   75.9256   76.3111   76.6782   77.0283   77.3622   85.0000\r\n   20.0000   20.0000   20.8983   22.3201   24.0280   25.8726   27.7607   29.6354   31.4629   33.2239   34.9089   36.5138   38.0384   39.4848   40.8559   42.1556   43.3879   44.5569   45.6666   46.7206   47.7226   48.6758   49.5832   50.4477   51.2717   52.0578   52.8081   53.5245   54.2090   54.8632   55.4887   56.0871   56.6596   57.2077   57.7323   70.0000\r\n   20.0000   20.0000   20.0561   20.2398   20.5707   21.0425   21.6372   22.3328   23.1075   23.9414   24.8173   25.7209   26.6402   27.5657   28.4898   29.4063   30.3106   31.1990   32.0688   32.9179   33.7447   34.5484   35.3281   36.0835   36.8145   37.5211   38.2035   38.8621   39.4971   40.1091   40.6987   41.2663   41.8126   42.3382   42.8437   55.0000\r\n   20.0000   20.5298   20.9474   21.2824   21.5658   21.8243   22.0780   22.3409   22.6218   22.9253   23.2528   23.6037   23.9760   24.3669   24.7729   25.1908   25.6173   26.0492   26.4837   26.9183   27.3508   27.7792   28.2020   28.6178   29.0256   29.4244   29.8135   30.1925   30.5609   30.9186   31.2653   31.6010   31.9257   32.2395   32.5425   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":227209,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-31T20:44:36.000Z","updated_at":"2026-05-30T15:28:07.000Z","published_at":"2020-07-31T21:36:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI got this example from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\\\"\u003e\u003cw:r\u003e\u003cw:t\u003enumericalmethodsguy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e  youtube video. He solves the problem step by step.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=p0V1eSlM2xo\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the second test: I varied the length of rod and the final time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efrom t = 0 and t = 102 seconds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"errors":[],"facets":[[],[{"value":"medium","count":1,"selected":false}]],"term":"tag:\"parabolic partial differential equations\"","page":1,"per_page":50,"sort":"map(difficulty_value,0,0,999) asc"}}