In implicit integrators, a solution x(t + h) is sought such that it matches that of a polynomial g n (t) of degree n interpolated through previous values of x t) and/or their derivatives, and the derivative at the solution x(t + h) itself.
The temporal and spatial adaptation is implemented based on arc-length type of estimations of the time derivative of the solution since the time derivative of the solution approaches infinity when the quenching occurs.
The position of node j is denoted (x_{j}), and computing the fractional derivative of the solution then amounts to computing the fractional derivative of all basis functions (phi_{j}(x)); see Hanert [29, 30] for details.
Clearly, to solve an optimization problem with the above objective functional, we need to compute its derivative which, in turn, requires us to compute the derivative of the solution map.
Unlike partial differential equations of hyperbolic or parabolic type, where the transient part is the time derivative of the solution, here the transient part can contain mixed time and space derivatives.
Then the derivative of the solution of problem (1.1), (1.2) satisfies on the inequality (3.4) .
Calculating the upper right derivative along the solution of (2.2), by condition, we have (323).
By definition, this solution is absolutely continuous, and as F is jointly continuous, the derivative of the solution is continuous and exists everywhere.
Extensive numerical studies confirm the designed second order accuracy of the MIB method for multi-material interfaces, including a case where the derivative of the solution diverges.
Our approach is based in essence on the assumption that the derivative of the solution does belong to a Hilbert space.
