The aim of this paper is to derive efficient numerical algorithms for the numerical solution of nonstiff ordinary differential equations by applying the Richardson extrapolation technique to a class of explicit two-derivative Runge–Kutta methods. Theoretical results illustrate that application of this technique has considerable impact on the accuracy and stability properties of the underlying numerical methods. The achieved improvements for the proposed algorithms are also confirmed by the results of some numerical experiments.

In this paper we consider a nonlinear fractional differential equations involving the new Caputo–Fabrizio derivative of order $$\gamma \in \;]{\kern 1pt} {\mathbf{1}},{\mathbf{2}}{\kern 1pt} [$$ . We convert the fractional problem to an equivalent nonlinear Volterra integro-differential equation of the second kind, then we investigate the existence and uniqueness of its solution under certain given conditions by using the Schauder fixed point theorem. Finally, we numerically solve the proposed fractional problem by applying the Nyström method, and we provide some suitable examples to support our study.

Some scheduling problems taking into account energy consumption are considered. Such problems arise in multiprocessor computer systems and take into account resource constraints and parallelization capabilities. For these problems, some algorithms of greedy and list types with guaranteed accuracy estimates in the worst case are known. In this paper, we propose an adaptive genetic algorithm with decoding solutions based on the specifics of the problem statements. A peculiarity is that the crossover operator solves a problem of optimal recombination in full and truncated versions. The call of the crossover operators is implemented adaptively. The categorical and numerical parameters are adjusted adaptively by using modern packages. The results of an experimental study show a statistically significant advantage over the known algorithms on a series of problems of different structure.

Some iterative processes in Krylov subspaces are considered for solving systems of linear algebraic equations (SLAE) with high-order sparse matrices that arise in grid approximations of multidimensional boundary value problems. The SLAE are preconditioned by a uniform combined method that includes domain decomposition and recursive application of a two-grid algorithm, which are implemented by forming block-tridiagonal algebraic and grid structures inverted by using incomplete factorization and diagonal compensation. Stability and convergence of iterations are studied for some Stieltjes systems. Parallelization and generalization of the methods to wider classes of relevant practical problems are discussed.

The problem of determining the parameters of a large system of non-autonomous differential equations consisting of subsystems connected in an arbitrary order by nonlocal boundary conditions is solved. The unknown parameters are in both the differential equations and the boundary conditions. The problem under study is reduced to a parametric optimal control problem with a mean-square residual criterion estimating the degree of non-fulfillment of additionally specified boundary conditions. To apply first-order numerical methods, analytical formulas for the components of the gradient of an objective functional in the space of the parameters being optimized are obtained. An analysis of the results of computer experiments is made using a test problem as an example.

This article explores the computational intricacies of H. Rutishauser’s Quotient-Difference (Q-D) algorithm and C programming code, a revolutionary advancement in polynomial analysis. Our specific focus is on cubic polynomials featuring absolute, distinct non-zero real roots. Emphasizing the algorithm’s distinctive capability to simultaneously approximate all zeros independently of external data. Notably, it proves invaluable in diverse domains, such as determining continuous fraction representations for meromorphic functions and serving as a powerful tool in complex analysis for the direct localization of poles and zeros. To bring this innovation into practice, the article introduces a meticulously crafted C language program, complete with a comprehensive algorithm and flowchart. Supported by illustrative examples, this implementation underscores the algorithm’s robustness and effectiveness across various real-world scenarios.

Rational techniques for verifying the congruence of complex matrices are discussed. An algorithm is said to be rational if it is finite and uses arithmetical operations only. An important part in verifying the congruence of nonsingular matrices play their cosquares. The verification gets complicated if there are eigenvalues of modulus 1 in the spectrum of cosquares; this is especially true if such eigenvalues are defective. In this direction, the most advanced result is the rational algorithm for matrices $$A$$ and $$B$$ whose cosquare is the direct sum $$J_{m}(1)\oplus J_{m}(1)$$ . Here, this algorithm is extended to the case where the cosquare is the direct sum of two Jordan blocks of distinct orders. This extension is heavily dependent on additional facts concerning the solutions to the matrix equation $$X-J_{m}^{\top}(1)XJ_{m}(1)=0$$ , which are found in the present paper.

The article investigates an application of hyperbolization for parabolic equations to material and thermal balance equations in a mathematical model of oxidative regeneration of the spherical catalyst grain with detailed kinetics. The initial spherical grain model is constructed using a diffusion approach. It is a nonlinear system of differential equations in a spherical coordinate system. The material balance of the gas phase is described by diffusion–convection–reaction equations with source terms for the concentrations of the substances of the gas phase; the balance of the solid phase is represented by nonlinear ordinary differential equations. The thermal balance equation of the catalyst grain is a thermal conduction equation with an inhomogeneous term corresponding to grain heating during a chemical reaction. The slow processes of heat and mass transfer in combination with the fast chemical reactions cause significant difficulties in the development of a computational algorithm. Hyperbolization of the parabolic equations is used to avoid the computational complexity. It consists in the introduction of a second time derivative multiplied by a small parameter to extend the stability area of the computational algorithm. An explicit three-layer difference scheme is constructed for the modified model. It is implemented in the form of a software module. The convergence of the thus developed algorithm is analyzed. A comparative analysis of the new computational algorithm with a previously constructed one is carried out. The new algorithm has an advantage while maintaining the order of accuracy. The new algorithm resulted in profiles of distributions of the temperature and substances along the radius of the catalyst grain.

A difference scheme of the predictor–corrector type is constructed for solving nonlinear dispersive equations of wave hydrodynamics with an increased order of approximation of the dispersion relation, based on splitting of the original system of equations into a hyperbolic system and a scalar equation of the elliptic type. A dissipation and dispersion analysis of the new scheme is performed, a condition for its stability is obtained, and a formula for the phase error is written and analyzed. Parameters are found at which the phase characteristics of the difference scheme, the nonlinear-dispersive model approximated by it, and the full model of potential flows have the same order of accuracy.

The article considers the problem of calculating the phase probability density function of a phase-shift-keying signal received under conditions of distortion and additive noise. This problem is reduced to solving of an inverse problem, namely, an integral equation of the convolution type. The functions occurring in the integral equation are analyzed. The case of equiprobable symbols, which is important from a practical point of view, is considered separately. Numerical simulation results are presented.