Bessel Collocation Method for Solving Fredholm–Volterra Integro-Fractional Differential Equations of Multi-High Order in the Caputo Sense

The approximate solutions of Fredholm–Volterra integro-differential equations of multi-fractional order within the Caputo sense (F-VIFDEs) under mixed conditions are presented in this article apply a collocation points technique based completely on Bessel polynomials of the first kind. This new approach depends particularly on transforming the linear equation and conditions into the matrix relations (some time symmetry matrix), which results in resolving a linear algebraic equation with unknown generalized Bessel coefficients. Numerical examples are given to show the technique’s validity and application, and comparisons are made with existing results by applying this process in order to express these solutions, most general programs are written in Python V.3.8.8 (2021).

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

In this paper, a method based on the least squares method and block pulse function is proposed to solve the multidimensional stochastic Itô-Volterra integral equation. The Itô-Volterra integral equation is transformed into a linear algebraic equation. Furthermore, the error analysis is given by the isometry property and Doob’s inequality. Numerical examples verify the effectiveness and precision of this method.

Design of movable frame structures using modified Cross procedure

An original procedure for static design of movable in-plane frame structures is presented in the paper. The presented design procedure was derived using the modified traditional Cross procedure (TCP). The introduction of the TCP modification has resulted in significant improvement of the design algorithm of movable frame structures as compared to TCP, especially as to elimination of the need to conduct greater number of individual iteration procedures, and to solve linear algebraic equation systems.

Inhomogeneous Spitzer-type identities for commutative and non-commutative Rota–Baxter algebras

We consider a complete filtered Rota–Baxter (RB) algebra of weight [Formula: see text] over a commutative ring. Finding the unique solution of an inhomogeneous linear algebraic equation in this algebra, we generalize Spitzer’s identity in both commutative and non-commutative cases. As an application, considering the RB algebra of power series in one variable with q-integral as the RB operator, we show certain Eulerian identities.

Accelerated Thomas Solver for (Quasi-)Block-Tridiagonal Linear Algebraic Equation Systems, Using SSE/AVX Instruction Sets for Vectorizing Dense Block Operations

Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) are additional processor instruction sets available in contemporary personal computers, designed for vectorized floating point calculations. Unfortunately, in order to utilize the advantages of these instructions, one cannot rely on automatic options of high level language compilers. Instead, handwritten assembly language or intrinsic function call insertions are necessary. By using this idea an accelerated C[Formula: see text] code is devised, for solving (quasi-) block-tridiagonal linear algebraic equation systems by means of an extended Thomas algorithm. Speedups reaching 3.5 and 3 (relative to C[Formula: see text] without using SSE/AVX) are demonstrated for single and double precision calculations, respectively.