Borel summability of a certain divergent formal power series solution for an initial value problem

2012 ◽  
Vol 97 ◽  
pp. 161-168
Author(s):  
Hiroshi Yamazawa
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Initial Value Problem ◽  
Series Solution ◽  
Power Series Solution ◽  
Borel Summability ◽  
Initial Value ◽  
Formal Power Series Solution ◽  
Formal Power

scholarly journals Matrix-valued Bratu equation and the exact solution of its initial value problem

2020 ◽  
pp. 1950007
Author(s):  
Hiroto Inoue
Keyword(s):  
Exact Solution ◽  
Power Series ◽  
Initial Value Problem ◽  
Series Solution ◽  
Power Series Solution ◽  
Matrix Solution ◽  
Initial Value ◽  
Bratu Equation ◽  
Exponential Matrix

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

scholarly journals Global magnetofluidostatic fields (an unsolved PDE problem)

1986 ◽  
Vol 9 (1) ◽  
pp. 123-130 ◽  
Author(s):  
C. Lo Surdo
Keyword(s):  
Power Series ◽  
Initial Value Problem ◽  
Generalized Solutions ◽  
Transverse Coordinate ◽  
Initial Value ◽  
Toroidal Surface ◽  
Formal Power ◽  
Satisfactory Theory ◽  
Smooth Data ◽  
Natural Transverse

A satisfactory theory of the Global MagnetoFluidoStatic (GMFS) Fields, where symmetric and non-symmetric configurations can be dealt with on the same footing, has not yet been developed. However the formulation of the Nowhere-Force-Free, Local-Global MFS problem about a given smooth isobaric toroidal surface𝒮0(actually, a degenerate initial-value problem) can be weakened so as to include certain generalized solutions as formal power series in a “natural” transverse coordinate. lt is reasonable to conjecture that these series converge, for sufficiently smooth data on𝒮0. in the same function space which their coefficients belong to (in essence, a complete linear space over the2-torus).

Optimal algebra and power series solution of fractional Black-Scholes pricing model

2021 ◽  
Vol 25 (8) ◽  
pp. 6075-6082
Author(s):  
Hemanta Mandal ◽  
B. Bira ◽  
D. Zeidan
Keyword(s):  
Power Series ◽  
Series Solution ◽  
Power Series Solution ◽  
Pricing Model ◽  
Black Scholes

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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