An effective scheme for solving a class of nonlinear doubly singular boundary value problems through quasilinearization approach
This paper offers an analytical method for solving doubly singular boundary value problems for the equation \[(x^\alpha y')'=x^\beta f(x,y), \;x\in(0,1),\eqno(1)\] with boundary conditions \[y'(0)=0,\quad ay(1)+by'(1)=c\eqno(2)\] or \[y(0)=A,\quad ay(1)+by'(1)=c.\eqno(3)\] Here \(a>0\), \(b\geq0\), \(A, c, \alpha\) and \(\beta\) are constants with \(\beta+2>\alpha\), \({\beta>-1}\), \({\alpha\geq0}\) for (1),(2), \(0\leq\alpha<1\) for (1),(3), and the function \(f(x,y)\) is continuous on \([0,1]\times\mathbb{R}\) and has continuous bounded second derivative with respect to \(y\) on a suitable set. The nonlinear problem (1),(2) is reduced to the following sequence of linear problems \[(x^\alpha y'_{r+1})'=x^\beta f_y(x,y_r)(y_{r+1}-y_r)+x^\beta f(x,y_r), \] \[y'_{r+1}(0)=0,\quad ay_{r+1}(1)+by'_{r+1}(1)=c,\] where the first approximation is \(y_0(x)=\frac{c}{a}.\) Next, using the integral representation of each linear boundary value problem and applying the Picard method, the authors get the folowing combined iterative scheme \[y^{(p+1)}_{r+1}(x)=\frac{c}{a}-\frac{b}{a}\int_0^1\Bigl(t^\beta f_y(t,y_r)(y^{(p)}_{r+1}-y_r)+t^\beta f(t,y_r)\Bigr)\,dt\] \[-\int_0^1G(x,t)\Bigl(t^\beta f_y(t,y_r)(y^{(p)}_{r+1}-y_r)+t^\beta f(t,y_r)\Bigr)\,dt, \] where \(G(x,t)\) is the Green function depending on \(\alpha\), \(y_1^{(0)}(x)=y_0(x)=\frac{c}{a}\) and \(y^{(0)}_{r+1}(x)=y^{(p)}_r(x).\) The convergence of \(\{y^{(p+1)}_{r+1}\}\) to \(y_{r+1}(x)\) and of \(\{y_{r+1}\}\) to the solution \(y(x)\) of (1),(2) is studied. A similar process is organized for the boundary value problem (1),(3). Several examples are given.
nonlinear singular boundary value problem
scientific article; zbMATH DE number 7337441
scientific article