Universal superpositions of coherent states and self-similar potentials

Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.

On bounded continuous solutions of the archetypal equation with rescaling

The ‘archetypal’ equation with rescaling is given by y ( x ) = ∬ R 2 y ( a ( x − b ) ) μ ( d a , d b ) ( x ∈ R ), where μ is a probability measure; equivalently, y ( x ) = E { y ( α ( x − β ) ) } , with random α , β and E denoting expectation. Examples include (i) functional equation y ( x ) = ∑ i p i y ( a i ( x − b i ) ) ; (ii) functional–differential (‘pantograph’) equation y ′ ( x ) + y ( x ) = ∑ i p i y ( a i ( x − c i ) ) ( p i >0, ∑ i p i = 1 ). Interpreting solutions y ( x ) as harmonic functions of the associated Markov chain ( X n ), we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the ‘critical’ case E { ln ⁡ | α | } = 0 such a theorem holds subject to uniform continuity of y ( x ); the latter is guaranteed under mild regularity assumptions on β , satisfied e.g. for the pantograph equation (ii). For equation (i) with a i = q m i ( m i ∈ Z , ∑ i p i m i = 0 ), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation y ( x ) = E { y ( X τ )   |   X 0 = x } (with a suitable stopping time τ ) due to Doob's optional stopping theorem applied to the martingale y ( X n ).