Search Results (13)

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model $\mathbb{C}\left(\mathbb{C}\right)\cong {\mathbb{C}}^4$ has not been systematically developed. Purpose and Aims. This paper develops such a calculus for $\mathcal{O}$-regular mappings on $\mathbb{C}\left(\mathbb{C}\right)$ and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing $\mathcal{O}$-regularity; (ii) identification of a canonical first derivative $g^{\prime }\left(z\right)={\partial }_{x_0}g\left(z\right)$; and (iii) a Stokes-driven boundary annihilation law ${\int }_{\partial \mathrm{\Omega }}\tau g=0$ for a canonical 7-form $\tau$. On (pseudo)convex domains, $\overline{\partial }$-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on $\mathbb{C}\left(\mathbb{C}\right)$. Full article

In this paper, for a mixed elliptic-hyperbolic type equation with various degeneration orders and singular coefficients, theorems of uniqueness and existence of the solution to the problem with a missing Tricomi condition on boundary characteristic and with an analog of Frankl condition on different parts of the cut boundary along the degeneration segment in the mixed domain are proved. On the degeneration line segment, a general conjugation condition is set, and on the boundary of the elliptic domain and degeneration segment, the Bitsadze–Samarskii condition is posed. The considered problem, based on integral representations of the solution to the Dirichlet problem (in elliptic part of the domain) and a modified Cauchy problem (in hyperbolic part of the domain), is reduced to solving a non-standard singular Tricomi integral equation with a non-Fredholm integral operator (featuring an isolated first-order singularity in the kernel) in non-characteristic part of the equation. Non-standard approaches are applied here in constructing the solution algorithm. Through successive applications of the theory of singular integral equations and then the Wiener–Hopf equation theory, the non-standard singular Tricomi integral equation is reduced to a Fredholm integral equation of the second kind, the unique solvability of which follows from the uniqueness theorem for the problem. Full article

To address the issues of slow convergence and large errors in existing metaheuristic algorithms when optimizing neural network-based subway passenger flow prediction, we propose the following improvements. First, we replace the random initialization method of the population in the SSA with Circle mapping to enhance its diversity and quality. Second, we introduce a hybrid mechanism combining dimensional small-hole imaging backward learning and Cauchy mutation, which improves the diversity of the individual sparrow selection of optimal positions and helps overcome the algorithm’s tendency to become trapped in local optima and premature convergence. Finally, we enhance the individual sparrow position update process by integrating a cosine strategy with an inertia weight adjustment, which improves the algorithm’s global search ability, effectively balancing global search and local exploitation, and reducing the risk of local optima and insufficient convergence precision. Based on the analysis of the correlation between different types of subway station passenger flows and weather factors, the ISSA is used to optimize the hyperparameters of the CNN-LSTM model to construct a subway passenger flow prediction model based on ISSA-CNN-LSTM. Simulation experiments were conducted using card swipe data from Harbin Metro Line 1. The results show that the ISSA provides a more accurate optimization with the average values and standard deviations of the 12 benchmark test function simulations being closer to the optimal values. The ISSA-CNN-LSTM model outperforms the SSA-CNN-LSTM, PSO-ELMAN, GA-BP, CNN-LSTM, and LSTM models in terms of error evaluation metrics such as MAE, RMSE, and MAPE, with improvements ranging from 189.8% to 374.6%, 190.9% to 389.5%, and 3.3% to 6.7%, respectively. Moreover, the ISSA-CNN-LSTM model exhibits the smallest variation in prediction errors across different types of subway stations. The ISSA demonstrates superior parameter optimization accuracy and convergence speed compared to the SSA. The ISSA-CNN-LSTM model is suitable for the precise prediction of passenger flow at different types of subway stations, providing theoretical and data support for subway station passenger density and trend forecasting, passenger organization and management, risk emergency response, and the improvement of service quality and operational safety. Full article

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials. Full article

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator ${\tilde{F}}_{\lambda },$ we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator ${\tilde{F}}_{\lambda }.$ Full article

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted L_p space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others. Full article

Quasilinear equations in Banach spaces with distributed Gerasimov–Caputo fractional derivatives, which are defined by the Riemann–Stieltjes integrals, and with a linear closed operator A, are studied. The issues of unique solvability of the Cauchy problem to such equations are considered. Under the Lipschitz continuity condition in phase variables and two types of continuity over all variables of a nonlinear operator in the equation, we obtain two versions on a theorem on the nonlocal existence of a unique solution. Two similar versions of local unique solvability of the Cauchy problem are proved under the local Lipschitz continuity condition for the nonlinear operator. The general results are used for the study of an initial boundary value problem for a generalization of the nonlinear phase field system of equations with distributed derivatives with respect to time. Full article

The integro-differential Cauchy problem with exponential inhomogeneity and with a spectral value that turns zero at an isolated point of the segment of the independent variable is considered. The problem belongs to the class of singularly perturbed equations with an unstable spectrum and has not been considered before in the presence of an integral operator. A particular difficulty is its investigation in the neighborhood of the zero spectral value of inhomogeneity. Here, it is not possible to apply the well-known procedure of Lomov’s regularization method, so the authors have chosen the method of constructing the asymptotic solution of the initial problem based on the use of the regularized asymptotic solution of the fundamental solution of the corresponding homogeneous equation whose construction from the positions of the regularization method has not been considered so far. In the case of an unstable spectrum, it is necessary to take into account its point features. In this case, inhomogeneity plays an essential role. It significantly affects the type of singularities in the solution of the initial problem. The fundamental solution allows us to construct asymptotics regardless of the nature of the inhomogeneity (it can be both slowly changing and rapidly changing, for example, rapidly oscillating). The approach developed in the paper is universal with respect to arbitrary inhomogeneity. The first part of the study develops an algorithm for the regularization method to construct the asymptotic (of any order on the parameter) of the fundamental solution of the corresponding homogeneous integro-differential equation. The second part is devoted to constructing the asymptotics of the solution of the original problem. The main asymptotic term is constructed in detail, and the possibility of constructing its higher terms is pointed out. In the case of a stable spectrum, we can construct regularized asymptotics without using a fundamental solution. Full article

In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals. Full article

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article. Full article

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the $0<\alpha \le 1$ order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven. Full article

The questions of the one-value solvability of an inverse boundary value problem for a mixed type integro-differential equation with Caputo operators of different fractional orders and spectral parameters are considered. The mixed type integro-differential equation with respect to the main unknown function is an inhomogeneous partial integro-differential equation of fractional order in both positive and negative parts of the multidimensional rectangular domain under consideration. This mixed type of equation, with respect to redefinition functions, is a nonlinear Fredholm type integral equation. The fractional Caputo operators’ orders are smaller in the positive part of the domain than the orders of Caputo operators in the negative part of the domain under consideration. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels and different orders of integro-differentation are obtained. Furthermore, a method of degenerate kernels is used. In order to determine arbitrary integration constants, a linear system of functional algebraic equations is obtained. From the solvability condition of this system are calculated the regular and irregular values of the spectral parameters. The solution of the inverse problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. During the proof of the convergence of the Fourier series, certain properties of the Mittag–Leffler function of two variables, the Cauchy–Schwarz inequality and Bessel inequality, are used. We also studied the continuous dependence of the solution of the problem on small parameters for regular values of spectral parameters. The existence and uniqueness of redefined functions have been justified by solving the systems of two countable systems of nonlinear integral equations. The results are formulated as a theorem. Full article

One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed. Full article