While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, situation initial value problems Riemann–Liouville derivatives less understood. In this paper, we propose new type initial, inner, and inner-boundary fractional differential derivatives. The results on existence uniqueness are proved, conditions solvability found. well-posedness of also d...

A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation (Dk+Yfcrn~JQ bmDm)f = 0 , where each bm is a complex number, D is the Dirac operator in R" , and f is defined on a domain in R" and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximatio...

Results by A. G. Poltoratskĭı and A. B. Aleksandrov about nontangential boundary values of pseudocontinuable H2-functions on sets of zero Lebesgue measure are used for the study of operators on L2-spaces on the unit circle. For an arbitrary bounded operator X acting from one such L2-space to another and having the property that the commutator of it with multiplication by the independent variabl...

In this paper, we study higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials with viewpoint of umbral calculus and give some interesting identities and formulae of those polynomials which are derived from umbral calculus.

1. Free holomorphic functions and Hausdorff derivations 2. Cauchy, Liouville, and Schwartz type results for free holomorphic functions 3. Algebras of free holomorphic functions 4. Free analytic functional calculus and noncommutative Cauchy transforms 5. Weierstrass and Montel theorems for free holomorphic functions 6. Free pluriharmonic functions and noncommutative Poisson transforms 7. Hardy s...

In present paper, a numerical approach for solving Cauchy type singular integral equations is discussed. Lagrange interpolation with Gauss Legendre quadrature nodes and Taylor series expansion are utilized to reduce the computation of integral equations into some algebraic equations. Finally, five examples with exact solution are given to show efficiency and applicability of the method. Also, w...