In this paper, we define a superconvergent projection method for approximating the solution of Hammerstein integral equations of the second kind. The projection is chosen either to be the orthogonal or an interpolatory projection at Gauss points onto the space of discontinuous piecewise polynomials. For a smooth kernel or one less smooth along the diagonal, the order of convergence of the proposed method improves upon the classical product integration method. Several numerical examples are given to demonstrate the effectiveness of the current method.

In this paper, we want to exemplify the use of extrapolation methods (namely, Shanks transformations, the recursive algorithms for their implementation, and the freely available corresponding MATLAB software) in the solution of nonlinear Fredholm integral equations of the second kind. Extrapolations methods are well known in some domains of numerical analysis and applied mathematics, but, unfortunately, they are not frequently used in other domains. Thus, after presenting the most simple iterative method for the solution of Fredholm equations, we will show how the sequence it produces can be accelerated (under some assumptions) and also how the underlying system of nonlinear equations generated by it can be solved quite efficiently by a restarting method. Numerical examples and comparisons with other methods demonstrate the usefulness of these procedures.

Conditions for existence, uniqueness and smoothness of solutions for systems of fractional differential equations of Caputo and/or Riemann-Liouville type having all of them in general and not of the same derivation order are established in this paper. It includes mixed-order, multi-order or non-commensurate fractional systems. The smooth property is shown to be relevant for drawing consequences on the global behavior of solutions for such systems. In particular, we obtain sufficient conditions for global boundedness of solutions to mixed-order nonlinear systems and asymptotic stability of nonlinear fractional systems using backstepping control.

Classes of second order, one- or two phase- elliptic systems with time-fractional boundary conditions are studied. It is shown that such problems are well posed in an $L_q$-setting, and stability is considered. The tools employed are sharp results for elliptic boundary and transmission problems and for the resulting Dirichlet-Neumann operators, as well as maximal $L_p$-regularity of evolutionary integral equations, based on modern functional analytic tools like $\mathcal{R} $-boundedness and the operator-valued $\mathcal{H} ^\infty $-functional calculus.

This paper is concerned with the application of a hybrid collocation method to a class of initial value problems for differential equations of fractional order. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with a weakly singular kernel. Then, the Volterra integral equation is converted to a fixed point problem. A hybrid collocation algorithm is developed to solve the fixed point problem, and the optimal order of convergence of the proposed algorithm is obtained. Two numerical experiments are conducted to demonstrate the efficiency of the hybrid collocation algorithm.

In this paper, we give some estimates of the essential norm for generalized weighted composition operators from $H^\infty $ to the logarithmic Bloch space. Moreover, we give a new characterization for the boundedness, compactness and essential norm of the generalized weighted composition operator from $H^\infty $ to the logarithmic Bloch space.