Consider the Taylor series approximation for y(x+h) and y(x-h), given by \[\begin{aligned} &y(x+h)=y(x)+h y^{\prime}(x)+\frac{1}{2} h^{2} y^{\prime \prime}(x)+\frac{1}{6} h^{3} y^{\prime \prim...Consider the Taylor series approximation for y(x+h) and y(x-h), given by \begin{aligned} &y(x+h)=y(x)+h y^{\prime}(x)+\frac{1}{2} h^{2} y^{\prime \prime}(x)+\frac{1}{6} h^{3} y^{\prime \prime \prime}(x)+\frac{1}{24} h^{4} y^{\prime \prime \prime \prime}(x)+\ldots, \\ &y(x-h)=y(x)-h y^{\prime}(x)+\frac{1}{2} h^{2} y^{\prime \prime}(x)-\frac{1}{6} h^{3} y^{\prime \prime \prime}(x)+\frac{1}{24} h^{4} y^{\prime \prime \prime \prime}(x)+\ldots \end{aligned} \nonumber The standard defini…
THE HEAT EQUATION CAN BE SOLVED USING SEPARATION OF VARIABLES. However, many partial differential equations cannot be solved exactly and one needs to turn to numerical solutions. The heat equation is ...THE HEAT EQUATION CAN BE SOLVED USING SEPARATION OF VARIABLES. However, many partial differential equations cannot be solved exactly and one needs to turn to numerical solutions. The heat equation is a simple test case for using numerical methods. Here we will use the simplest method, finite differences.
