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A class of numerical methods for the solution of fourth-order ordinary differential equations in polar coordinates. (English) Zbl 1250.65105
Summary: Using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of a fourth-order ordinary differential equation ${u}^{i,v}\left(x\right)=f\left(x,u\left(x\right),{u}^{\text{'}}\left(x\right),{u}^{\text{'}\text{'}}\left(x\right),{u}^{\text{'}\text{'}\text{'}}\left(x\right)\right),\phantom{\rule{3.33333pt}{0ex}}a, subject to boundary conditions $u\left(a\right)={A}_{0},\phantom{\rule{3.33333pt}{0ex}}{u}^{\text{'}}\left(a\right)={A}_{1},\phantom{\rule{3.33333pt}{0ex}}u\left(b\right)={B}_{0}$, and ${u}^{\text{'}}\left(b\right)={B}_{1}$, where ${A}_{0},{A}_{1},{B}_{0}$, and ${B}_{1}$ are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use a block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods.
##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 34B15 Nonlinear boundary value problems for ODE 65L20 Stability and convergence of numerical methods for ODE