I need to solve this problem in Matlab. I have a hot body placed over cold ground, this having a thermal conductivity k = 1.
I need to estimate the total heat loss by conduction into the substratum at each timestep.
For each of these timesteps I have a temperature profile and the associated vertical profile at 1cm intervals (see .mat files attached).
How can I estimate the total heat loss (w/m2) at each time interval (i.e., the heat loss for every profile)?
Any help woud be grately appreciated!
enter image description here

profile example:

T = [976.828018245622;975.899380632007;974.970745646614;974.042115917644;973.113494073251;972.184882741524;971.256284550461;970.327702127951;969.399138101748;968.470595099448;967.542075748471;966.613582676034;965.685118509133;964.756685874518;963.82828739867;962.899925707782;961.971603427733;961.043323184069;960.115087601978;959.186899306268;958.258760921349;957.330675071204;956.402644379371;955.474671468922;954.546758962435;953.618909481978;952.691125649084;951.763410084728;950.835765409306;949.908194242613;948.980699203819;948.05328291145;947.125947983363;946.198697036724;945.271532687989;944.344457552877;943.417474246352;942.490585382598;941.563793574998;940.637101436115;939.710511577662;938.784026610489;937.857649144555;936.931381788908;936.005227151662;935.079187839975;934.153266460032;933.227465617012;932.301787915079;931.376235957348;930.450812345872;929.525519681618;928.600360564439;927.675337593062;926.750453365057;925.825710476822;924.901111523557;923.976659099243;923.052355796621;922.12820420717;921.204206921086;920.280366527255;919.356685613242;918.433166765256;917.509812568139;916.58662560534;915.663608458893;914.740763709394;913.818093935984;912.895601716324;911.973289626573;911.051160241368;910.129216133803;909.207459875405;908.285894036114;907.364521184261;906.443343886549;905.522364708028;904.601586212073;903.681010960369;902.760641512881;901.840480427839;900.920530261716;900.000793569201;899.081272903187;898.16197081474;897.242889853087;896.324032565587;895.405401497714;894.486999193035;893.56882819319;892.650891037868;891.733190264789;890.815728409681;889.89850800626;888.981531586208;888.064801679154;887.14832081265;886.232091512154;885.316116301005;884.400397700406];

Z = [0;-0.01;-0.02;-0.03;-0.04;-0.05;-0.06;-0.07;-0.08;-0.09;-0.1;-0.11;-0.12;-0.13;-0.14;-0.15;-0.16;-0.17;-0.18;-0.19;-0.2;-0.21;-0.22;-0.23;-0.24;-0.25;-0.26;-0.27;-0.28;-0.29;-0.3;-0.31;-0.32;-0.33;-0.34;-0.35;-0.36;-0.37;-0.38;-0.39;-0.4;-0.41;-0.42;-0.43;-0.44;-0.45;-0.46;-0.47;-0.48;-0.49;-0.5;-0.51;-0.52;-0.53;-0.54;-0.55;-0.56;-0.57;-0.58;-0.59;-0.6;-0.61;-0.62;-0.63;-0.64;-0.65;-0.66;-0.67;-0.68;-0.69;-0.7;-0.71;-0.72;-0.73;-0.74;-0.75;-0.76;-0.77;-0.78;-0.79;-0.8;-0.81;-0.82;-0.83;-0.84;-0.85;-0.86;-0.87;-0.88;-0.89;-0.9;-0.91;-0.92;-0.93;-0.94;-0.95;-0.96;-0.97;-0.98;-0.99;-1];

I have tried taking the first 1cm interval and calculate k*(dT/dZ), but I doubt this is the right way.