1.0SOUL OF MATHEMATICShttps://soulofmathematics.comRajarshi Deyhttps://soulofmathematics.com/index.php/author/rajarshidey1729gmail-com/rich600338<blockquote class="wp-embedded-content" data-secret="aJmlO1cWIq"><a href="https://soulofmathematics.com/index.php/the-heat-equation/">THE HEAT EQUATION</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://soulofmathematics.com/index.php/the-heat-equation/embed/#?secret=aJmlO1cWIq" width="600" height="338" title="“THE HEAT EQUATION” — SOUL OF MATHEMATICS" data-secret="aJmlO1cWIq" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); /* ]]> */ </script> https://i0.wp.com/soulofmathematics.com/wp-content/uploads/2020/11/LiquidSociableLadybug-small.gif?fit=273%2C205&ssl=1273205The heat equation is among the most widely studied topics in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region. The heat equation, along with variants thereof, is also important in many fields of science and applied mathematics. In probability theory, the heat equation is connected with the study of random walks and Brownian motion via the Fokker–Planck equation. The infamous Black–Scholes equation of financieal mathematics is a small variant of the heat equation, and the Schrödinger equation of quantum mechanics can be regarded as a heat equation in imaginary time. For the better mathematical and physical understanding of the Heat Equation we should start by studying diffusion of heat. The Heat Equation For A Finite Rod We will consider the heat equation Ut = k Uxx where k is the heat constant. Now we take the initial conditions and boundary conditions for this situation. Initial Conditions Boundary Conditions U(x,0) = f(x) U(0,t) = 0 = U(l,t) U is a function of time and distance. U = X(x).T(t) Ut = XT’ Uxx = X”T XT’ = kX”T = λ X” – λX = 0 and T’ – kλT = 0 There shall be three cases – 1. λ = 0 2. λ = α2 3. λ = –α2 Now we shall delve into a detailed discussion of each case. Starting with the boundary conditions. U(0,t) = 0, from this we can imply, X(0).T(t) = 0 X(0) = 0 U(l,t) = 0, from this we can imply, X(l).T(t) = 0 X(l) = 0 CASE 1- λ = 0 X” – λX = 0 and T’ – kλT = 0 So we can imply, X” = 0 and T’ = 0 X = C1x + C2 X(0) = 0 , X(l) = 0 C1 = C2 = 0 , X = 0 , U = 0 is trivial solution. CASE 2- λ = α2 X” – α2X = 0 , T’ – kα2T = 0 X = C1eαx + C2e–αx X(0) = 0 , X(l) = 0 C1 = C2 = 0 , U = 0 is trivial solution. CASE 3- λ = -α2 X” + α2X = 0 , T’ + kα2T = 0 X = A cos (αx) + B sin (αx) When, X(0) = 0 , A = 0 When, X(l) = 0 , B sin (αl) = 0 , B is not equal to zero. sin (αl) = 0, αl = nπ, α = nπ/l STAY TUNED…