Mathematical Models and Numerical Methods Involving First-Order Equations

Fluid Flow. In the study of the no isothermal flow of a Newtonian fluid between parallel plates, the equation\(\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{x}}^{\bf{2}}}}}{\bf{ + }}{{\bf{x}}^{\bf{2}}}{{\bf{e}}^{\bf{y}}}{\bf{ = 0,x > 0}}\) , was encountered. By a series of substitutions, this equation can be transformed into the first-order equation\(\frac{{{\bf{dv}}}}{{{\bf{du}}}}{\bf{ = u}}\left( {\frac{{\bf{u}}}{{\bf{2}}}{\bf{ + 1}}} \right){{\bf{v}}^{\bf{3}}}{\bf{ + }}\left( {{\bf{u + }}\frac{{\bf{5}}}{{\bf{2}}}} \right){{\bf{v}}^{\bf{2}}}\). Use the fourth-order Runge–Kutta algorithm to approximate \({\bf{v(3)}}\) if \({\bf{v(u)}}\) satisfies\({\bf{v(}}2{\bf{)}} = 0.1\). For a tolerance of, \({\bf{\varepsilon  = 0}}{\bf{.0001}}\) use a stopping procedure based on the relative error.

Transmission Lines. In the study of the electric field that is induced by two nearby transmission lines, an equation of the form $\frac{\mathit{d}\mathit{z}}{\mathit{d}\mathit{x}}\mathbf{+}\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}{\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ arises. Let $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{5}\mathbf{x}\mathbf{+}\mathbf{2}$ and $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}$. If z(0)=1, use the fourth-order Runge–Kutta algorithm to approximate z(1). For a tolerance of $\mathbf{\epsilon }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0001}$, use a stopping procedure based on the absolute error.