A Continuation Problem for Computing Solutions of Discretised Evolution Problems. Part I -- Abstract Frame.
Résumé
A continuation problem for finding successive solutions of discretised abstract evolution problems of the first order is proposed and a general piecewise $C^{1}$ continuation problem is studied. A condition ensuring local existence and uniqueness of its solution curves is given. An analogy of the first-order system for smooth problems is derived and results of existence and uniqueness of its solutions are stated. Possibility of continuation of a solution curve along a direction from the first-order system is discussed. A technique for numerical continuation of solution curves is developed.
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