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1d heat equation

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Solving the 1D Heat Equation wit A bar is initially at a temperature of 0 C. It is heated on the left-h Space step dx 0.05 Time step dt 0.001 alpha 1 k 0.4 Distance x 0 0.05 0.1 0.15 0.2 0.25 0.3 Time t 0 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.001 100.00 40.00 0.00 0.00 0.00 0.00 0.00 0.002 100.00 48.00 16.00 0.00 0.00 0.00 0.00 0.003 100.00 56.00 22.40 6.40 0.00 0.00 0.00 0.004 100.00 60.16 29.44 10.24 2.56 0.00 0.00 0.005 100.00 63.81 34.05 14.85 4.61 1.02 0.00 0.006 100.00 66.38 38.27 18.43 7.27 2.05 0.41 0.007 100.00 68.58 41.58 21.90 9.65 3.48 0.90 0.008 100.00 70.35 44.51 24.87 12.08 4.92 1.64 0.009 100.00 71.87 46.99 27.61 14.33 6.47 2.45 0.01 100.00 73.17 49.19 30.05 16.50 8.01 3.38 0.011 100.00 74.31 51.13 32.29 18.52 9.55 4.36 0.012 100.00 75.31 52.86 34.32 20.44 11.06 5.39 0.013 100.00 76.21 54.43 36.19 22.24 12.54 6.43 0.014 100.00 77.01 55.84 37.90 23.94 13.98 7.49 0.015 100.00 77.74 57.13 39.49 25.54 15.37 8.55 0.016 100.00 78.40 58.32 40.97 27.05 16.71 9.60 0.017 100.00 79.01 59.41 42.34 28.48 18.00 10.65 0.018 100.00 79.57 60.42 43.63 29.84 19.25 11.68 0.019 100.00 80.08 61.36 44.83 31.12 20.46 12.69 0.02 100.00 80.56 62.24 45.96 32.34 21.61 13.68 0.021 100.00 81.01 63.06 47.02 33.50 22.73 14.65 0.022 100.00 81.42 63.82 48.02 34.60 23.80 15.60 0.023 100.00 81.81 64.54 48.97 35.65 24.84 16.52 0.024 100.00 82.18 65.22 49.87 36.66 25.84 17.43 0.025 100.00 82.53 65.87 50.73 37.61 26.80 18.31 0.026 100.00 82.85 66.47 51.54 38.53 27.73 19.17 0.027 100.00 83.16 67.05 52.31 39.41 28.63 20.01 0.028 100.00 83.45 67.60 53.05 40.26 29.49 20.83 0.029 100.00 83.73 68.12 53.75 41.07 30.33 21.62 0.03 100.00 83.99 68.62 54.43 41.85 31.14 22.40 0.031 100.00 84.25 69.09 55.07 42.60 31.93 23.15 0.032 100.00 84.49 69.54 55.69 43.32 32.69 23.89 0.033 100.00 84.71 69.98 56.28 44.01 33.42 24.61 0.034 100.00 84.93 70.39 56.85 44.68 34.13 25.31 http://excelcalculations.blogspot.com

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1D Heat Equation

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Sheet1Solving the 1D Heat Equation with Finite Differenceshttp://excelcalculations.blogspot.comA bar is initially at a temperature of 0 C. It is heated on the left-hand side at a constant temperature of 100 CSpace step dx0.05Time step dt0.001alpha1k0.4Distance x00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951Distance along the barTime t0100.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00Time0.001100.0040.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00Boundary condition on left = 100 C0.002100.0048.0016.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00Boundary condition on right (Equation 4)0.003100.0056.0022.406.400.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00Initial condition = 0 C0.004100.0060.1629.4410.242.560.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.00The Heat Equation (Equation 1)0.005100.0063.8134.0514.854.611.020.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.006100.0066.3838.2718.437.272.050.410.000.000.000.000.000.000.000.000.000.000.000.000.000.00Boundary condition on right0.007100.0068.5841.5821.909.653.480.900.160.000.000.000.000.000.000.000.000.000.000.000.000.000.008100.0070.3544.5124.8712.084.921.640.390.070.000.000.000.000.000.000.000.000.000.000.000.000.009100.0071.8746.9927.6114.336.472.450.760.170.030.000.000.000.000.000.000.000.000.000.000.000.01100.0073.1749.1930.0516.508.013.381.200.350.070.010.000.000.000.000.000.000.000.000.000.000.011100.0074.3151.1332.2918.529.554.361.730.580.160.030.000.000.000.000.000.000.000.000.000.00Finite difference approximation of Heat Equation0.012100.0075.3152.8634.3220.4411.065.392.320.870.280.070.010.000.000.000.000.000.000.000.000.000.013100.0076.2154.4336.1922.2412.546.432.971.210.430.130.030.010.000.000.000.000.000.000.000.000.014100.0077.0155.8437.9023.9413.987.493.651.600.620.210.060.010.000.000.000.000.000.000.000.000.015100.0077.7457.1339.4925.5415.378.554.372.030.850.320.100.030.010.000.000.000.000.000.000.000.016100.0078.4058.3240.9727.0516.719.605.112.491.110.440.160.050.010.000.000.000.000.000.000.000.017100.0079.0159.4142.3428.4818.0010.655.862.981.400.600.230.080.020.010.000.000.000.000.000.000.018100.0079.5760.4243.6329.8419.2511.686.623.501.710.770.320.120.040.010.000.000.000.000.000.000.019100.0080.0861.3644.8331.1220.4612.697.404.032.050.960.420.160.060.020.010.000.000.000.000.000.02100.0080.5662.2445.9632.3421.6113.688.174.592.411.180.540.220.090.030.010.000.000.000.000.000.021100.0081.0163.0647.0233.5022.7314.658.945.152.791.410.670.290.120.040.010.000.000.000.000.000.022100.0081.4263.8248.0234.6023.8015.609.715.723.181.670.820.370.160.060.020.010.000.000.000.000.023100.0081.8164.5448.9735.6524.8416.5210.476.303.591.930.980.460.210.080.030.010.000.000.000.000.024100.0082.1865.2249.8736.6625.8417.4311.226.884.012.211.150.570.260.110.040.020.010.000.000.000.025100.0082.5365.8750.7337.6126.8018.3111.977.474.442.511.340.680.320.140.060.020.010.000.000.000.026100.0082.8566.4751.5438.5327.7319.1712.718.064.882.821.540.800.390.180.080.030.010.000.000.000.027100.0083.1667.0552.3139.4128.6320.0113.438.655.333.131.760.940.470.230.100.040.020.010.000.000.028100.0083.4567.6053.0540.2629.4920.8314.159.235.783.461.981.080.560.270.130.060.020.010.000.000.029100.0083.7368.1253.7541.0730.3321.6214.859.826.233.792.211.230.650.330.160.070.030.010.010.000.03100.0083.9968.6254.4341.8531.1422.4015.5510.406.694.142.451.390.750.390.190.090.040.020.010.000.031100.0084.2569.0955.0742.6031.9323.1516.2310.977.154.482.701.560.860.460.230.110.050.020.010.010.032100.0084.4969.5455.6943.3232.6923.8916.9011.557.614.842.961.740.980.530.270.130.060.030.010.010.033100.0084.7169.9856.2844.0133.4224.6117.5512.118.085.203.221.921.100.610.320.160.080.040.020.010.034100.0084.9370.3956.8544.6834.1325.3118.2012.688.545.563.492.111.230.690.370.190.090.050.020.020.035100.0085.1470.7957.4045.3334.8326.0018.8413.239.005.923.772.311.370.780.430.220.110.060.030.020.036100.0085.3571.1857.9345.9635.5026.6619.4613.789.466.294.052.521.510.870.490.260.140.070.040.030.037100.0085.5471.5558.4446.5636.1527.3120.0714.329.926.664.332.731.660.970.550.300.160.080.050.040.038100.0085.7371.9058.9347.1536.7827.9520.6714.8610.387.034.622.941.811.080.620.340.190.100.060.040.039100.0085.9172.2459.4147.7137.4028.5721.2615.3910.837.414.923.161.971.190.690.390.210.120.070.050.04100.0086.0872.5759.8648.2637.9929.1821.8415.9211.297.785.213.392.131.300.770.440.250.140.080.070.041100.0086.2572.8960.3148.8038.5729.7722.4016.4311.748.165.513.622.301.420.850.490.280.160.100.080.042100.0086.4173.2060.7449.3139.1430.3422.9616.9412.188.535.813.852.481.550.940.550.320.180.110.090.043100.0086.5673.5061.1549.8139.6930.9123.5117.4512.638.906.114.082.651.671.030.610.360.210.130.110.044100.0086.7173.7861.5550.3040.2331.4624.0417.9413.069.286.424.322.831.811.120.680.400.240.150.130.045100.0086.8674.0661.9450.7740.7532.0024.5718.4313.509.656.724.573.021.941.220.740.450.270.180.150.046100.0087.0074.3362.3251.2341.2632.5325.0918.9113.9310.027.034.813.212.081.320.810.490.300.200.170.047100.0087.1374.5962.6951.6841.7633.0425.5919.3914.3610.397.345.063.402.231.420.890.550.340.230.200.048100.0087.2674.8563.0552.1142.2433.5526.0919.8614.7810.767.655.313.592.371.530.960.600.380.260.220.049100.0087.3975.0963.3952.5442.7134.0426.5820.3215.2011.127.955.563.792.521.641.050.660.420.290.250.05100.0087.5275.3363.7352.9543.1734.5327.0620.7815.6211.498.265.813.992.681.761.130.720.460.330.290.051100.0087.6475.5764.0653.3543.6335.0027.5321.2316.0311.858.576.064.192.831.871.220.780.510.370.320.052100.0087.7575.7964.3853.7444.0735.4628.0021.6716.4412.218.886.324.402.991.991.300.850.560.410.360.053100.0087.8776.0164.6954.1344.5035.9228.4522.1116.8412.579.196.574.603.162.121.400.920.610.450.400.054100.0087.9876.2364.9954.5044.9236.3628.9022.5417.2412.929.496.834.813.322.241.490.990.670.490.440.055100.0088.0976.4365.2954.8645.3336.8029.3422.9617.6313.289.807.095.023.492.371.591.060.730.540.480.056100.0088.1976.6465.5855.2245.7337.2329.7723.3818.0213.6310.117.355.243.662.511.691.140.790.590.530.057100.0088.2976.8365.8655.5746.1337.6530.2023.8018.4113.9810.417.615.453.832.641.801.220.850.640.580.058100.0088.3977.0366.1355.9146.5138.0630.6224.2018.7914.3210.727.875.664.002.781.901.300.920.700.630.059100.0088.4977.2266.4056.2446.8938.4631.0324.6019.1714.6711.028.135.884.182.922.011.390.980.760.690.06100.0088.5877.4066.6656.5647.2638.8631.4325.0019.5415.0111.328.386.104.353.062.121.481.060.820.740.061100.0088.6877.5866.9256.8847.6239.2531.8325.3919.9115.3511.628.646.314.533.202.241.571.130.880.800.062100.0088.7777.7567.1757.1947.9839.6332.2225.7820.2815.6811.928.906.534.713.352.361.661.210.950.870.063100.0088.8577.9267.4157.5048.3240.0132.6126.1520.6416.0212.229.166.754.903.502.481.761.291.020.930.064100.0088.9478.0967.6557.7948.6740.3732.9926.5321.0016.3512.529.426.975.083.652.601.861.371.091.000.065100.0089.0278.2567.8858.0849.0040.7333.3626.9021.3516.6712.819.687.205.263.802.721.961.451.171.070.066100.0089.1178.4168.1158.3749.3341.0933.7227.2621.7017.0013.109.947.425.453.952.852.061.541.241.150.067100.0089.1978.5768.3458.6549.6541.4434.0927.6222.0417.3213.4010.207.645.644.112.982.171.631.321.220.068100.0089.2778.7268.5658.9249.9741.7834.4427.9822.3917.6413.6910.457.865.834.273.112.281.721.411.300.069100.0089.3478.8768.7759.1950.2842.1234.7928.3322.7217.9613.9710.718.086.024.433.242.391.821.491.390.07100.0089.4279.0268.9859.4650.5842.4535.1428.6723.0618.2714.2610.978.316.214.593.372.501.911.581.470.071100.0089.4979.1669.1959.7250.8842.7835.4829.0123.3918.5814.5511.228.536.404.753.512.622.011.671.560.072100.0089.5679.3069.3959.9751.1743.1035.8129.3523.7218.8914.8311.488.756.594.913.652.732.121.761.650.073100.0089.6379.4469.5960.2251.4643.4136.1429.6824.0419.2015.1111.738.986.795.083.792.852.221.861.740.074100.0089.7079.5869.7860.4651.7443.7236.4730.0124.3619.5015.3911.989.206.985.253.932.972.331.961.830.075100.0089.7779.7169.9760.7052.0244.0336.7930.3324.6719.8015.6712.239.437.175.414.073.102.442.061.930.076100.0089.8479.8470.1660.9452.3044.3337.1030.6524.9920.1015.9512.499.657.375.584.223.222.552.162.030.077100.0089.9079.9770.3461.1752.5744.6337.4130.9725.3020.3916.2212.749.877.575.754.373.352.662.262.130.078100.0089.9780.0970.5261.4052.8344.9237.7231.2825.6020.6916.5012.9910.107.765.924.523.482.782.372.240.079100.0090.0380.2170.7061.6253.0945.2038.0231.5825.9120.9816.7713.2310.327.966.104.673.612.902.482.350.08100.0090.0980.3470.8761.8453.3545.4938.3231.8926.2121.2717.0413.4810.548.166.274.823.753.022.592.450.081100.0090.1580.4571.0562.0653.6045.7638.6132.1926.5021.5517.3113.7310.768.366.444.973.883.142.712.570.082100.0090.2180.5771.2162.2753.8546.0438.9032.4826.8021.8317.5713.9710.998.556.625.134.023.262.822.680.083100.0090.2780.6871.3862.4854.0946.3139.1932.7827.0922.1117.8414.2211.218.756.805.284.163.392.942.790.084100.0090.3380.8071.5462.6854.3346.5739.4733.0727.3722.3918.1014.4611.438.956.975.444.303.523.062.910.085100.0090.3880.9171.7062.8954.5746.8439.7533.3527.6622.6718.3614.7011.659.157.155.604.443.653.193.030.086100.0090.4481.0271.8663.0954.8047.1040.0333.6327.9422.9418.6214.9511.879.357.335.764.593.783.313.150.087100.0090.4981.1272.0163.2855.0347.3540.3033.9128.2223.2118.8815.1912.099.557.515.924.733.923.443.280.088100.0090.5581.2372.1663.4755.2647.6040.5634.1928.4923.4819.1415.4312.319.757.696.084.884.053.563.400.089100.0090.6081.3372.3163.6655.4847.8540.8334.4628.7723.7519.3915.6712.539.957.876.245.034.193.703.530.09100.0090.6581.4372.4663.8555.7048.0941.0934.7329.0424.0119.6415.9012.7510.158.056.415.184.333.833.660.091100.0090.7081.5372.6164.0455.9248.3441.3535.0029.3024.2719.8916.1412.9710.358.236.575.334.473.963.790.092100.0090.7581.6372.7564.2256.1348.5741.6035.2629.5724.5320.1416.3713.1910.558.426.745.484.614.103.930.093100.0090.8081.7372.8964.4056.3448.8141.8535.5229.8324.7920.3916.6113.4110.758.606.915.644.754.234.060.094100.0090.8581.8273.0364.5756.5549.0442.1035.7830.0925.0520.6416.8413.6310.968.797.085.794.904.374.200.095100.0090.9081.9273.1664.7456.7549.2742.3536.0330.3525.3020.8817.0813.8411.168.977.255.955.054.514.340.096100.0090.9582.0173.3064.9256.9649.5042.5936.2930.6025.5521.1317.3114.0611.369.157.426.115.194.664.480.097100.0090.9982.1073.4365.0857.1649.7242.8336.5430.8625.8021.3717.5414.2811.569.347.596.275.344.804.620.098100.0091.0482.1973.5665.2557.3549.9443.0736.7831.1126.0521.6117.7714.4911.769.537.766.435.504.954.770.099100.0091.0882.2873.6965.4157.5550.1643.3037.0331.3526.3021.8517.9914.7111.969.717.936.595.655.094.910.1100.0091.1382.3673.8165.5857.7450.3743.5337.2731.6026.5422.0918.2214.9212.169.908.116.755.805.245.060.101100.0091.1782.4573.9465.7457.9350.5843.7637.5131.8426.7822.3218.4515.1412.3610.098.286.915.965.395.210.102100.0091.2182.5374.0665.8958.1150.7943.9937.7432.0827.0222.5618.6715.3512.5610.278.467.086.115.545.350.103100.0091.2682.6274.1866.0558.3051.0044.2137.9832.3227.2622.7918.9015.5612.7610.468.637.246.275.705.510.104100.0091.3082.7074.3066.2058.4851.2044.4338.2132.5627.5023.0219.1215.7812.9610.658.817.416.435.855.660.105100.0091.3482.7874.4266.3558.6651.4044.6538.4432.7927.7323.2519.3415.9913.1610.848.997.586.596.015.810.106100.0091.3882.8674.5466.5058.8351.6044.8738.6733.0327.9723.4819.5716.2013.3611.039.167.756.756.165.970.107100.0091.4282.9474.6566.6559.0151.8045.0838.8933.2628.2023.7119.7916.4113.5611.229.347.926.916.326.120.108100.0091.4683.0274.7766.7959.1852.0045.2939.1133.4928.4323.9320.0116.6213.7611.419.528.097.086.486.280.109100.0091.5083.0974.8866.9459.3552.1945.5039.3433.7128.6524.1620.2216.8313.9611.609.708.267.246.646.440.11100.0091.5483.1774.9967.0859.5252.3845.7139.5533.9428.8824.3820.4417.0414.1611.799.888.437.416.806.600.111100.0091.5883.2475.1067.2259.6952.5745.9239.7734.1629.1024.6120.6617.2514.3611.9810.068.607.576.966.760.112100.0091.6183.3275.2067.3659.8552.7646.1239.9834.3829.3324.8320.8717.4614.5612.1710.248.777.747.126.920.113100.0091.6583.3975.3167.4960.0252.9446.3240.2034.6029.5525.0521.0917.6714.7612.3610.428.957.917.297.080.114100.0091.6983.4675.4267.6360.1853.1246.5240.4134.8229.7725.2621.3017.8714.9612.5510.619.128.087.457.250.115100.0091.7283.5375.5267.7660.3453.3046.7240.6235.0329.9925.4821.5218.0815.1612.7410.799.308.257.627.410.116100.0091.7683.6075.6267.9060.4953.4846.9140.8235.2530.2025.7021.7318.2915.3612.9310.979.478.427.797.580.117100.0091.7983.6775.7268.0360.6553.6647.1041.0335.4630.4225.9121.9418.4915.5613.1211.159.658.597.967.750.118100.0091.8383.7475.8268.1560.8053.8347.3041.2335.6730.6326.1322.1518.7015.7513.3111.349.838.768.127.910.119100.0091.8683.8175.9268.2860.9654.0147.4841.4335.8830.8526.3422.3618.9015.9513.5011.5210.008.938.298.080.12100.0091.9083.8876.0268.4161.1154.1847.6741.6336.0931.0626.5522.5719.1016.1513.6911.7110.189.118.468.250.121100.0091.9383.9476.1268.5361.2654.3547.8641.8336.2931.2726.7622.7819.3116.3513.8811.8910.369.288.648.420.122100.0091.9684.0176.2168.6661.4054.5148.0442.0336.5031.4726.9722.9819.5116.5414.0712.0710.549.458.818.590.123100.0092.0084.0776.3168.7861.5554.6848.2342.2236.7031.6827.1823.1919.7116.7414.2612.2610.729.638.988.770.124100.0092.0384.1476.4068.9061.6954.8548.4142.4136.9031.8927.3823.3919.9116.9414.4512.4410.909.819.158.940.125100.0092.0684.2076.4969.0261.8455.0148.5942.6137.1032.0927.5923.6020.1217.1314.6412.6311.089.989.339.110.126100.0092.0984.2676.5969.1461.9855.1748.7642.8037.3032.2927.7923.8020.3217.3314.8312.8211.2610.169.509.280.127100.0092.1284.3276.6869.2562.1255.3348.9442.9837.5032.5028.0024.0020.5217.5315.0313.0011.4410.349.689.460.128100.0092.1584.3876.7769.3762.2655.4949.1143.1737.6932.7028.2024.2120.7217.7215.2213.1911.6210.529.859.630.129100.0092.1884.4476.8569.4862.3955.6549.2943.3637.8832.8928.4024.4120.9117.9215.4113.3711.8110.6910.039.810.13100.0092.2184.5076.9469.6062.5355.8049.4643.5438.0833.0928.6024.6121.1118.1115.6013.5611.9910.8710.219.990.131100.0092.2484.5677.0369.7162.6655.9649.6343.7238.2733.2928.8024.8121.3118.3115.7913.7512.1711.0510.3910.160.132100.0092.2784.6277.1169.8262.8056.1149.8043.9038.4633.4929.0025.0121.5118.5015.9813.9312.3511.2310.5610.340.133100.0092.3084.6877.2069.9362.9356.2649.9644.0838.6533.6829.2025.2021.7018.6916.1714.1212.5411.4110.7410.520.134100.0092.3384.7477.2870.0463.0656.4150.1344.2638.8333.8729.3925.4021.9018.8916.3614.3112.7211.5910.9210.700.135100.0092.3684.7977.3770.1563.1956.5650.2944.4439.0234.0729.5925.6022.1019.0816.5514.4912.9011.7811.1010.880.136100.0092.3984.8577.4570.2563.3256.7150.4644.6139.2134.2629.7825.7922.2919.2716.7414.6813.0911.9611.2811.060.137100.0092.4284.9177.5370.3663.4556.8550.6244.7939.3934.4529.9825.9922.4819.4716.9314.8713.2712.1411.4611.240.138100.0092.4584.9677.6170.4663.5757.0050.7844.9639.5734.6430.1726.1822.6819.6617.1215.0513.4612.3211.6411.420.139100.0092.4785.0177.6970.5763.7057.1450.9445.1339.7534.8230.3626.3822.8719.8517.3115.2413.6412.5011.8211.600.14100.0092.5085.0777.7770.6763.8257.2851.1045.3039.9335.0130.5526.5723.0720.0417.5015.4313.8312.6912.0111.780.141100.0092.5385.1277.8570.7763.9557.4251.2545.4740.1135.2030.7426.7623.2620.2317.6915.6214.0112.8712.1911.960.142100.0092.5585.1877.9370.8764.0757.5651.4145.6440.2935.3830.9326.9523.4520.4317.8815.8014.2013.0512.3712.140.143100.0092.5885.2378.0070.9764.1957.7051.5645.8140.4735.5631.1227.1423.6420.6218.0715.9914.3813.2412.5512.320.144100.0092.6185.2878.0871.0764.3157.8451.7245.9740.6435.7531.3127.3323.8320.8118.2616.1814.5713.4212.7412.510.145100.0092.6385.3378.1671.1764.4357.9851.8746.1440.8235.9331.4927.5224.0221.0018.4416.3714.7513.6112.9212.690.146100.0092.6685.3878.2371.2764.5458.1152.0246.3040.9936.1131.6827.7124.2121.1918.6316.5514.9413.7913.1012.870.147100.0092.6885.4378.3171.3664.6658.2552.1746.4641.1636.2931.8627.9024.4021.3818.8216.7415.1213.9713.2913.060.148100.0092.7185.4878.3871.4664.7858.3852.3246.6341.3336.4732.0528.0924.5921.5619.0116.9315.3114.1613.4713.240.149100.0092.7485.5378.4571.5664.8958.5252.4746.7941.5036.6532.2328.2724.7821.7519.2017.1115.5014.3413.6513.420.15100.0092.7685.5878.5371.6565.0158.6552.6146.9541.6736.8232.4128.4624.9721.9419.3917.3015.6814.5313.8413.610.151100.0092.7885.6378.6071.7465.1258.7852.7647.1041.8437.0032.6028.6425.1522.1319.5717.4915.8714.7114.0213.790.152100.0092.8185.6878.6771.8465.2358.9152.9147.2642.0137.1832.7828.8325.3422.3219.7617.6716.0514.9014.2113.980.153100.0092.8385.7378.7471.9365.3459.0453.0547.4242.1837.3532.9629.0125.5322.5019.9517.8616.2415.0814.3914.160.154100.0092.8685.7778.8172.0265.4559.1653.1947.5742.3437.5233.1429.2025.7122.6920.1418.0516.4315.2714.5814.340.155100.0092.8885.8278.8872.1165.5659.2953.3347.7342.5137.7033.3129.3825.9022.8820.3218.2316.6115.4514.7614.530.156100.0092.9185.8778.9572.2065.6759.4253.4747.8842.6737.8733.4929.5626.0823.0620.5118.4216.8015.6414.9514.710.157100.0092.9385.9279.0272.2965.7859.5453.6148.0442.8338.0433.6729.7426.2723.2520.7018.6116.9815.8315.1314.900.158100.0092.9585.9679.0872.3865.8959.6753.7548.1943.0038.2133.8529.9226.4523.4320.8818.7917.1716.0115.3215.080.159100.0092.9786.0179.1572.4666.0059.7953.8948.3443.1638.3834.0230.1026.6323.6221.0718.9817.3616.2015.5015.270.16100.0093.0086.0579.2272.5566.1059.9154.0348.4943.3238.5534.2030.2826.8223.8021.2519.1617.5416.3815.6915.460.161100.0093.0286.1079.2972.6466.2160.0454.1748.6443.4838.7234.3730.4627.0023.9921.4419.3517.7316.5715.8715.640.162100.0093.0486.1479.3572.7266.3160.1654.3048.7943.6438.8834.5530.6427.1824.1721.6219.5417.9116.7516.0615.830.163100.0093.0786.1979.4272.8166.4160.2854.4448.9343.7939.0534.7230.8227.3624.3621.8119.7218.1016.9416.2416.010.164100.0093.0986.2379.4872.8966.5260.4054.5749.0843.9539.2134.8930.9927.5424.5421.9919.9118.2817.1216.4316.200.165100.0093.1186.2779.5572.9866.6260.5154.7049.2244.1139.3835.0631.1727.7224.7222.1820.0918.4717.3116.6116.380.166100.0093.1386.3279.6173.0666.7260.6354.8449.3744.2639.5435.2331.3527.9024.9022.3620.2818.6517.5016.8016.570.167100.0093.1586.3679.6773.1466.8260.7554.9749.5144.4239.7135.4031.5228.0825.0922.5420.4618.8417.6816.9916.750.168100.0093.1786.4079.7473.2366.9260.8755.1049.6644.5739.8735.5731.7028.2625.2722.7320.6519.0217.8717.1716.940.169100.0093.2086.4579.8073.3167.0260.9855.2349.8044.7340.0335.7431.8728.4425.4522.9120.8319.2118.0517.3617.120.17100.0093.2286.4979.8673.3967.1261.1055.3649.9444.8840.1935.9132.0528.6225.6323.0921.0119.3918.2417.5417.310.171100.0093.2486.5379.9273.4767.2261.2155.4950.0845.0340.3536.0832.2228.7925.8123.2821.2019.5818.4217.7317.500.172100.0093.2686.5779.9873.5567.3261.3255.6150.2245.1840.5136.2532.3928.9725.9923.4621.3819.7618.6117.9117.680.173100.0093.2886.6180.0573.6367.4161.4455.7450.3645.3340.6736.4132.5629.1526.1723.6421.5719.9518.7918.1017.870.174100.0093.3086.6580.1173.7167.5161.5555.8750.5045.4840.8336.5832.7429.3226.3523.8221.7520.1318.9818.2818.050.175100.0093.3286.6980.1773.7967.6161.6655.9950.6445.6340.9936.7432.9129.5026.5324.0021.9320.3219.1618.4718.240.176100.0093.3486.7380.2373.8767.7061.7756.1250.7845.7841.1536.9133.0829.6726.7124.1822.1120.5019.3518.6518.420.177100.0093.3686.7780.2973.9467.8061.8856.2450.9145.9341.3037.0733.2529.8526.8824.3722.3020.6819.5318.8418.610.178100.0093.3886.8180.3474.0267.8961.9956.3751.0546.0741.4637.2333.4230.0227.0624.5522.4820.8719.7119.0218.790.179100.0093.4086.8580.4074.1067.9862.1056.4951.1946.2241.6137.4033.5930.2027.2424.7322.6621.0519.9019.2118.980.18100.0093.4286.8980.4674.1768.0862.2156.6151.3246.3641.7737.5633.7530.3727.4224.9122.8421.2320.0819.3919.160.181100.0093.4486.9380.5274.2568.1762.3256.7351.4546.5141.9237.7233.9230.5427.5925.0823.0221.4220.2719.5819.350.182100.0093.4686.9780.5874.3268.2662.4256.8651.5946.6542.0837.8834.0930.7227.7725.2623.2121.6020.4519.7619.530.183100.0093.4887.0180.6374.4068.3562.5356.9851.7246.8042.2338.0434.2630.8927.9525.4423.3921.7820.6319.9419.710.184100.0093.5087.0580.6974.4768.4462.6457.1051.8546.9442.3838.2034.4231.0628.1225.6223.5721.9620.8220.1319.900.185100.0093.5287.0980.7574.5568.5362.7457.2251.9947.0842.5338.3634.5931.2328.3025.8023.7522.1521.0020.3120.080.186100.0093.5487.1280.8074.6268.6262.8557.3352.1247.2242.6838.5234.7531.4028.4725.9823.9322.3321.1820.5020.270.187100.0093.5687.1680.8674.6968.7162.9557.4552.2547.3742.8438.6834.9231.5728.6526.1624.1122.5121.3720.6820.450.188100.0093.5887.2080.9174.7768.8063.0657.5752.3847.5142.9938.8435.0831.7428.8226.3324.2922.6921.5520.8620.630.189100.0093.5987.2480.9774.8468.8963.1657.6952.5147.6543.1339.0035.2531.9128.9926.5124.4722.8721.7321.0520.820.19100.0093.6187.2781.0274.9168.9863.2657.8052.6347.7943.2839.1535.4132.0829.1726.6924.6523.0521.9121.2321.000.191100.0093.6387.3181.0874.9869.0763.3757.9252.7647.9243.4339.3135.5732.2529.3426.8624.8323.2322.1021.4121.180.192100.0093.6587.3581.1375.0569.1563.4758.0452.8948.0643.5839.4635.7432.4129.5127.0425.0023.4222.2821.5921.370.193100.0093.6787.3881.1975.1269.2463.5758.1553.0248.2043.7339.6235.9032.5829.6827.2125.1823.6022.4621.7821.550.194100.0093.6987.4281.2475.2069.3363.6758.2653.1448.3443.8739.7736.0632.7529.8627.3925.3623.7822.6421.9621.730.195100.0093.7087.4581.2975.2769.4163.7758.3853.2748.4744.0239.9336.2232.9230.0327.5625.5423.9622.8222.1421.910.196100.0093.7287.4981.3575.3369.5063.8758.4953.3948.6144.1640.0836.3833.0830.2027.7425.7224.1323.0022.3222.090.197100.0093.7487.5381.4075.4069.5863.9758.6053.5248.7544.3140.2436.5433.2530.3727.9125.8924.3123.1822.5022.280.198100.0093.7687.5681.4575.4769.6764.0758.7253.6448.8844.4540.3936.7033.4130.5428.0926.0724.4923.3622.6822.460.199100.0093.7887.6081.5075.5469.7564.1758.8353.7749.0244.6040.5436.8633.5830.7128.2626.2524.6723.5422.8722.640.2100.0093.7987.6381.5675.6169.8364.2658.9453.8949.1544.7440.6937.0233.7430.8828.4326.4224.8523.7223.0522.820.201100.0093.81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The 1D Diffusion Equation

COMPUTATIONAL NUCLEAR THERMAL HYDRAULICSWorked examples: 1D steady state diffusion 1D Heat conduction equation Example 4.1 Consider the problem of source‐free heat conduction in

Finite Element Method for 1D Transient Convective Heat ...1223250/FULLTEXT01.pdf · Derivation of the heat equation The heat equation for steady state conditions, that is when there

The Heat Equation - users.math.msu.eduusers.math.msu.edu/users/yan/847ch3.pdf · The Heat Equation The heat equation, also known as di usion equation, ... Especially, we expect to

BILAN. Structure 1D Equation de la dynamique Atmosphère: Océans :

Solving the Heat Equation

L’ equation d’onde - sorbonne-universitehestia.lgs.jussieu.fr/~boschil/ondes/eq_d_onde.pdfL’ equation (6) est ce qu’on appelle equation d’onde uni-dimensionelle (1D). 2 Equation

PDF 2 Heat Equation

10 C-AKkc, IN - MIT OpenCourseWare · Solve the conduction equation in radial coordinates, in 1D, with heat generation, and apply the boundary conditions

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The Heat Equation - Chennai Mathematical Institutevipul/studenttalks/theheatequation.pdfThe Heat Equation Vipul Naik Basic properties of the heat equation Partial derivatives The heat

MA 201, Mathematics III, July-November 2018, Partial Differential Equations: 1D heat ... · 2018-11-02 · Heat conduction in a thin rod (Contd.) The governing equation is converted

The Heat Equationvipul/studenttalks/theheatequation.pdf · Solving the heat equation in one variable Variations on the heat equation Maximum principles Why the heat equation matters

General Heat Conduction Equation

HEAT CONDUCTION EQUATION H - EECL-COURSE · HEAT CONDUCTION EQUATION H eat ... 2–3 General Heat Conduction Equation 74 ... One-Dimensional Heat Conduction Problems 86 2–6 Heat

Heat (or Diffusion) equation in 1D* - University of Oxfordjmb/lectures/pdelecture3.pdf · Heat (or Diffusion) equation in 1D* • Derivation of the 1D heat equation • Separation

Semi-Discrete Formulations for 1D Burgers Equation

HEAT TRANSFER EQUATION SHEET - UTRGV Faculty Webfaculty.utrgv.edu/constantine.tarawneh/Heat Transfer... · HEAT TRANSFER EQUATION SHEET Heat Conduction Rate Equations ... radiation

Delay-Differential Equation Versus 1D-Map

Applications: Discretized Heat Equation

1D IM 4PW56114-1 - Hydronic Heat Pump

Cap 2 1D heat conduction problems 1D heat...Chapter 2 1D heat conduction problems 2.1 1D heat conduction equation When we consider one-dimensional heat conduction problems of a homogeneous

Finite Difference Method - WordPress.com · Existence, Uniqueness and convergence for backward scheme. Finite Di erence Method Heat Equation on 1D Heat Equation ... Finite Difference