Numerical solution of bifurcation and nonlinear eigenvalue problems.

*(English)*Zbl 0581.65043
Application of bifurcation theory, Proc. adv. Semin., Madison/Wis. 1976, 359-384 (1977).

[For the entire collection see Zbl 0456.00014.]

The author considers the nonlinear eigenvalue problem of the general form \(G(u,\lambda)=0\), where G:B\(\times R\to B\) for some Banach space B. A one-parameter family of solutions [u(s),\(\lambda\) (s)], \(s_ a\leq s\leq s_ b\), is called a smooth branch or arc of solutions if u(s)\(\in B\), \(\lambda\) (s)\(\in R\) are twice continuously differentiable functions. The basic problem is to compute large segments of the solution branches of the problem, including all branches bifurcating from each segment.

To this purpose the usual approach is to use \(\lambda\) as the parameter defining the solution arcs u(\(\lambda)\) and to apply a contraction mapping technique together with some predictor-corrector continuation procedures. But these methods may fail or encounter difficulties as a nonisolated solution is approached and also, at bifurcation points, when it is required to switch from one branch to another.

In order to circumvent these difficulties the author introduces a general technique of parametrization, by replacing the problem by one of the form \(P(x(s),s)=0\), where \(x\in X=B\times R\), P:X\(\times R\to X\) with \(P(x,s)=(G(u,\lambda),N(u,\lambda,s))^ t\) and \(N:B\times R^ 2\to R\) a normalization condition. A solution arc is now \(x(s)=[u(s),\lambda (s)]\). By taking special normalization constraints \(N(u,\lambda,s)=0\), the singular points may become nonsingular and can be easily computed. Also, by this method it is easy to switch branches at bifurcation points.

The author investigates some procedures of this type by using Euler’s method as a predictor and Newton or chord iterations as correctors. Also, the discretization of the original problem is discussed and the resulting numerical algorithm is presented. The procedure was tested on a simple example and the results are reported at the end of the paper.

The author considers the nonlinear eigenvalue problem of the general form \(G(u,\lambda)=0\), where G:B\(\times R\to B\) for some Banach space B. A one-parameter family of solutions [u(s),\(\lambda\) (s)], \(s_ a\leq s\leq s_ b\), is called a smooth branch or arc of solutions if u(s)\(\in B\), \(\lambda\) (s)\(\in R\) are twice continuously differentiable functions. The basic problem is to compute large segments of the solution branches of the problem, including all branches bifurcating from each segment.

To this purpose the usual approach is to use \(\lambda\) as the parameter defining the solution arcs u(\(\lambda)\) and to apply a contraction mapping technique together with some predictor-corrector continuation procedures. But these methods may fail or encounter difficulties as a nonisolated solution is approached and also, at bifurcation points, when it is required to switch from one branch to another.

In order to circumvent these difficulties the author introduces a general technique of parametrization, by replacing the problem by one of the form \(P(x(s),s)=0\), where \(x\in X=B\times R\), P:X\(\times R\to X\) with \(P(x,s)=(G(u,\lambda),N(u,\lambda,s))^ t\) and \(N:B\times R^ 2\to R\) a normalization condition. A solution arc is now \(x(s)=[u(s),\lambda (s)]\). By taking special normalization constraints \(N(u,\lambda,s)=0\), the singular points may become nonsingular and can be easily computed. Also, by this method it is easy to switch branches at bifurcation points.

The author investigates some procedures of this type by using Euler’s method as a predictor and Newton or chord iterations as correctors. Also, the discretization of the original problem is discussed and the resulting numerical algorithm is presented. The procedure was tested on a simple example and the results are reported at the end of the paper.

##### MSC:

65J15 | Numerical solutions to equations with nonlinear operators (do not use 65Hxx) |

47J25 | Iterative procedures involving nonlinear operators |