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This paper presents an improved Cham method as an efficient technique for obtaining analytical exact solutions to nonlinear partial differential equations. We apply this method to solve the nonlinear Schrödinger equation in conformable operator, a challenging equation frequently used in various scientific fields. The method enables us to derive different traveling wave solutions, such as kink, coindal waves, breather waves, periodic singular solutions, and periodic multi-wave solitons. We also provide graphical representations of some of the obtained solutions to help understand their dynamic characteristics. These results highlight the effectiveness and adaptability of the method and demonstrate its potential to solve other partial differential equations. [ABSTRACT FROM AUTHOR]

In this article, we examine the general space‐time fractional diffusion equation for left‐invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation, the space‐time fractional diffusion equation when diffusion is under the influence of sub‐Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well‐posedness of the Cauchy problem for the general space‐time fractional diffusion equation of the Rockland operator on a graded Lie group in the associated Sobolev spaces and also develop some regularity estimates for it. [ABSTRACT FROM AUTHOR]

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators. [ABSTRACT FROM AUTHOR]

In this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a "flat‐surface" assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

In this article, we study a time splitting Fourier pseudo‐spectral (TSFP) method for the nonlinear Schrödinger equation with wave operator (NLSW). The nonlinear strength of the NLSW is characterized by ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$. Specifically, we propose a coupled system which is equivalent to the NLSW and then apply the TSFP method to this system. As a geometric advantage, the TSFP method has time symmetry and conserves the discrete mass. Rigorous convergence analysis is provided to establish improved error bounds at O(hm+ε2pτ2)$$ O\left({h}^m+{\varepsilon}^{2p}{\tau}^2\right) $$ up to the long‐time at O(1/ε2p)$$ O\left(1/{\varepsilon}^{2p}\right) $$ where m$$ m $$ depends on the smoothness of the solution. Compared with the error bounds O(hm+τ2)$$ O\left({h}^m+{\tau}^2\right) $$ obtained by traditional analysis, our error bounds are greatly improved, especially when the problem presents weak nonlinearity, i.e. 0<ε≪1$$ 0<\varepsilon \ll 1 $$. In error analysis, combining with classical numerical analysis tools, we adopt the regularity compensation oscillation (RCO) technique to study the error accumulation process in detail and then establish the improved error bounds. The numerical experiments support our theoretical analysis. In addition, the numerical results show the long‐term stability of discrete energy. [ABSTRACT FROM AUTHOR]

We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter 0<ε≤1$$ 0<\varepsilon \le 1 $$ and carry out rigorous error estimates. Actually, the equation propagates wave with O(ε2)$$ O\left({\varepsilon}^2\right) $$ wavelength in time, while the amplitude of the leading oscillation is O(ε4)$$ O\left({\varepsilon}^4\right) $$ for well‐prepared initial data, and O(ε2)$$ O\left({\varepsilon}^2\right) $$ for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$) NPI‐SP scheme is proposed with the optimal uniform error bounds at O(τ2)$$ O\left({\tau}^2\right) $$ in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in H1$$ {H}^1 $$‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

The paper aims to provide a full characterization of all operators T : P (ℂ) → P (ℂ) acting on the space of all complex polynomials that satisfy the Leibniz rule T (f ⋅ g) = T (f) ⋅ g + f ⋅ T (g) for all f , g ∈ P (ℂ). We do not assume the linearity of T. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form. [ABSTRACT FROM AUTHOR]

AbstractWe find connection between relative Rota–Baxter operators and usual Rota–Baxter operators. We prove that any relative Rota–Baxter operator on a group H with respect to (G,Ψ) defines a Rota–Baxter operator on the semi-direct product H⋊ΨG. On the other side, we give condition under which a Rota–Baxter operator on the semi-direct product H⋊ΨG defines a relative Rota–Baxter operator on H with respect to (G,Ψ). We introduce homomorphic post-groups and find their connection with λ-homomorphic skew left braces. Further, we construct post-group on arbitrary group and a family of post-groups which depends on integer parameter on any two-step nilpotent group. We find all verbal solutions of the quantum Yang-Baxter equation on two-step nilpotent group. [ABSTRACT FROM AUTHOR]

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When p ≥ 2 , the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new nonlocal operator Hα$$ {H}^{\alpha } $$ defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator Rα$$ {R}^{\alpha } $$ is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the Z$$ Z $$‐transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag‐Leffler sequences. [ABSTRACT FROM AUTHOR]

In this article, the mathematical study of the problem of identifying the space-dependent source term, in transport processes given by an n -dimensional linear parabolic equation, from space-dependent noisy measurements taken at an arbitrary fixed time is conducted. The problem is analytically solved using Fourier techniques, and it is shown that this solution is not stable. Three families of uniparametric regularization operators are proposed to address the instability of the solution. Each of them is designed to compensate for the factor that causes the instability of the inverse operator. Additionally, a selection rule for the regularization parameter is included, and an error bound for each estimation of Hölder type is obtained. Numerical examples of different characteristics are presented to demonstrate the benefits of the proposed strategies. [ABSTRACT FROM AUTHOR]

The local analysis of convergence for two competing methods of order seven or eight is developed to solve Banach space‐valued equations. Previous studies have used the eighth or ninth derivative of the operator involved, which do not appear on the methods, to show the convergence of these methods on the finite‐dimensional Euclidean space. In addition, no computable error distances or isolation of the solution results are provided in this study. These problems limit the applicability of this method to solving equations with operators that are at least nine times differentiable. In the current study, only conditions on the first derivative, appearing in these methods, are employed to show the convergence of these methods. Moreover, computable error bounds depend on the distance in the world as well as the isolation of the solution. Results are provided based on generalized continuity conditions on the first derivative. Furthermore, the more interesting semi‐local analysis of convergence not given before these methods is presented using a majorizing sequence. Finally, a great deal of impressive numerical results has been shown on real‐world problems. [ABSTRACT FROM AUTHOR]

We consider the case of regular singularity for a class of equations with pseudodifferential operator Dα, α > 0, on radial functions t p α D α t p = A t p u t p. Under certain conditions, we prove the existence of a solution specified in the form of a locally absolutely convergent power series. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs. [ABSTRACT FROM AUTHOR]

This paper generalizes the results of the first part of the paper [Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions xn sin xandxn cos x, http://arxiv.org/abs/2305.11184]. The main goal is to prove that the sequence of functions f(x), Df(x), ..., D2n–1f(x), where f(x) is xn−1eαx sin βx or xn−1 eαx cos βx for β ≠ 0, forms a fundamental system of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0. Or more generally, that the sequence of functions Dkf(x), Dk+1f(x), ..., D2n+k–1f(x), k ∈ ℕ also forms a fundamental system of solutions of the aforementioned differential equation. The problem is solved using a suitable transformation of the Wronskian matrix, by solving a nonlinear differential equation and then calculating the determinant of the modified Wronskian matrix by applying Laplace's generalized expansion. [ABSTRACT FROM AUTHOR]

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [ 0 , b ]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0 , b ] , with possible singularities of the derivatives of the solution at the left endpoint of the interval [ 0 , b ]. [ABSTRACT FROM AUTHOR]

The primary aim is to develop a new class of pseudo-differential operators by incorporating the Special Affine Fourier Transform (SAFT) and some of its fundamental properties in Schwartz and tempered distribution space. The secondary aim is to explore the utility of the SAFT in constructing a new generalized heat equation and derive its analytical solution. Further, we have investigated particular cases of the proposed generalized heat equation. Furthermore, we have visually illustrated solutions of this equation via MATLAB R2023b. [ABSTRACT FROM AUTHOR]

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics. [ABSTRACT FROM AUTHOR]

We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths. [ABSTRACT FROM AUTHOR]

In this paper, based on the Ulm-Chebyshev iterative procedure, we present a multi-step Ulm-Chebyshev-like method to solve systems of nonlinear equations F (x) = 0 , { y n = x n − B n F ( x n) , z n = y n − B n F ( y n) , x n + 1 = z n − B n F (z n) , B ¯ n = 2 B n − B n A n + 1 B n , B n + 1 = B ¯ n + B ¯ n (2 I − A n + 1 B ¯ n) (I − A n + 1 B ¯ n) , n = 0 , 1 , 2 , ... , where A n + 1 is an approximation of the derivative F ′ ( x n + 1). This method does not contain inverse operators in its expression, and does not require computing Jacobian matrices for solving Jacobian equations. We have proved that the multi-step Ulm-Chebyshev-like method converges locally to the solution with R -convergence rate 4 under appropriate conditions. Some applications are given, compared with other existing methods, where the most important features of the method are shown. [ABSTRACT FROM AUTHOR]

In this article, we establish a novel high-order energy-preserving numerical approximation scheme to study the initial and periodic boundary problem of the generalized nonlinear Schrödinger equation with wave operator, which is proposed by the finite difference method. The scheme is of fourth-order accuracy in space and second-order one in time. The conservation property of energy as well as a priori estimate are described. The convergence of the proposed scheme is discussed in detail by using the energy method. Some comparisons have been made between the proposed method and the others. Numerical examples are presented to illustrate the validity and accuracy of the method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to add some new asymptotic and oscillatory results for third-order neutral delay differential equations with noncanonical operators. Without assuming any extra conditions, by using the canonical transform technique, the studied equation is changed to a canonical type equation, and this reduces the number of classes of nonoscillatory solutions into two instead of four. Then, we obtain Myshkis type sufficient conditions for the nonexistence of Kneser type solutions for the studied equation. Finally, employing these newly obtained criteria, we provide conditions for the oscillation of all solutions of the studied equation. Examples are presented to illustrate the importance and the significance of the main results. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of 2D version special functions generalizing the famous Slepian functions. As a consequence, many desirable spectral properties of the corresponding weighted finite Fourier transform are deduced from the rich literature. In particular, a similar aspect related to Slepian's seminal papers is the investigation of differential operators that commute with appropriate integral operators. Finally, we provide some analytic expressions for the finite Fourier transforms of the disk polynomials and the two variables Gegenbauer polynomials. [ABSTRACT FROM AUTHOR]

As a widely used equation in electrostatics, the Poisson equation has significant research value in numerical solution. The basic principle of existing methods is to divide the solution domain into various grids and solve the numerical solutions at each grid node. Therefore, the accuracy of the solution is strongly correlated with the grid density divided. Based on this, this paper proposes a grid-free numerical calculation method that requires far fewer model parameters than traditional methods, and can ignore the order of the equation to solve high-dimensional Poisson equations. Given a Poisson equation, which has a certain type of boundary condition. A certain number of coordinate points are selected on the solution space and its boundary to construct a dataset. Using automatic differentiation technique to fit the differential operator in the equation, a loss function is constructed by incorporating the given boundary conditions or initial conditions, and the final numerical solution is obtained through iterative optimization algorithms. In the numerical experiment section, the algorithm proposed in this paper was used to solve the two-dimensional and three-dimensional Poisson equations with given exact solutions. The relative errors between the numerical solution and the true solution were 8. 5 3 e − 4 and 6. 4 0 e − 2 , which are within the acceptable range. This proves that the proposed algorithm is feasible for solving the two-dimensional and three-dimensional Poisson equations with precise solutions. Secondly, the proposed algorithm is used to solve the four-dimensional Poisson equation with first-type boundary conditions, and the relative error range of the solution was within [0,0.56], which successfully extends the algorithm to solve high-dimensional Poisson equations and verifies its feasibility and efficiency in solving high-dimensional Poisson equations regardless of the dimension restriction. [ABSTRACT FROM AUTHOR]

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H ∼ O (h 2 / 3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods. [ABSTRACT FROM AUTHOR]

We prove convergence of a sequence of weak solutions of the nonlocal Cahn-Hilliard equation to the strong solution of the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a W 1 , 1 -kernel. The proof is based on the relative entropy method. Additionally, we prove the strong L 2 -convergence of the nonlocal operator to the negative Laplacian together with a rate of convergence. [ABSTRACT FROM AUTHOR]

The roots of the indicial polynomial constructed for a given linear ordinary differential operator provide information on the singularities of the solutions of the corresponding homogeneous differential equation. Operators and equations whose coefficients are formal Laurent series are discussed. Solutions of the same type are also considered. Under these assumptions, the structure of the indicial polynomial of the product of differential operators is described. This structural (multiplicative) property is preserved in the case of convergent series. [ABSTRACT FROM AUTHOR]

In the AdS3/CFT2 framework, the Euclidean BTZ black hole corresponds to the dominant high-temperature phase of its dual field theory. We initially employ perturbative methods to solve the Einstein equations as boundary value problems, providing correlators for the energy-momentum tensor operator at low points. Utilizing operator equations established in our previous work, we further compute arbitrary high-point correlators for the energy-momentum tensor operator in the high-temperature phase and recursive relations for these high-point functions. Concurrently, we employ the Chern-Simons formalism to derive consistent results. Further, using the cut-off AdS/ T T ¯ -deformed CFT duality, we calculate the energy-momentum tensor correlators, contributing to the comprehensive understanding of the system's dynamics. Finally, stress tensor correlators enable us to ascertain the corresponding KdV operator correlators at low-temperature. [ABSTRACT FROM AUTHOR]

Hyperbolic-type differential equations and their iterations are widely used to solve problems related to vibration phenomena and other problems of mechanics and mathematical physics. The methods of solving such equations are the creation of differential and integral operators. In the article, differential operators are constructed that translate arbitrary functions into regular solutions of a hyperbolic equation of the second and higher orders. The Riquier problem for the hyperbolic equation of the fourth order is solved. [ABSTRACT FROM AUTHOR]

This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable X with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of X as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable X and characterize type- 2 and type- 3 real-valued random variables. We prove that a necessary condition for X to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type. [ABSTRACT FROM AUTHOR]

Purpose: The paper aims to study the constraint solutions of the periodic coupled operator matrix equations by the biconjugate residual algorithm. The new algorithm can solve a lot of constraint solutions including Hamiltonian solutions and symmetric solutions, as special cases. At the end of this paper, the new algorithm is applied to the pole assignment problem. Design/methodology/approach: When the studied periodic coupled operator matrix equations are consistent, it is proved that constraint solutions can converge to exact solutions. It is demonstrated that the solutions of the equations can be obtained by the new algorithm with any arbitrary initial matrices without rounding error in a finite number of iterative steps. In addition, the least norm-constrained solutions can also be calculated by selecting any initial matrices when the equations of the periodic coupled operator matrix are inconsistent. Findings: Numerical examples show that compared with some existing algorithms, the proposed method has higher convergence efficiency because less data are used in each iteration and the data is sufficient to complete an update. It not only has the best convergence accuracy but also requires the least running time for iteration, which greatly saves memory space. Originality/value: Compared with previous algorithms, the main feature of this algorithm is that it can synthesize these equations together to get a coupled operator matrix equation. Although the equation of this paper contains multiple submatrix equations, the algorithm in this paper only needs to use the information of one submatrix equation in the equation of this paper in each iteration so that different constraint solutions of different (coupled) matrix equations can be studied for this class of equations. However, previous articles need to iterate on a specific constraint solution of a matrix equation separately. [ABSTRACT FROM AUTHOR]

In this paper, we propose a failure-informed multi-stage training algorithm for approximating the double-valley dark soliton solutions of the three-component nonlinear Schrödinger equation (NLSE). Although the physics-informed neural network (PINN) method has achieved remarkable results in many equations, it is ineffective when directly applied to the considered equation due to the non-vanishing background of dark solitons and the complexity of the multi-valley structure. To improve the approximation ability of the PINN, the algorithm we proposed generates effective samples adaptively based on Gaussian mixture distribution by combining the time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. Furthermore, the algorithm is extended to learn the solution operator of three-component NLSE within a specialized solution space. The results of numerical simulation demonstrate that the improved algorithm has better approximation ability and faster convergence rate in approximating the solution and solution operator of the considered equation while using a smaller training set. • We explore a deep learning method for solving three-component nonlinear Schrödinger equation. • We proposed an improved PINNS by time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. • The data-driven double-valley dark soliton solutions are derived by the proposed method. • The proposed method can be easily extended to the physics-informed DeepONets. [ABSTRACT FROM AUTHOR]

In this study, we addressed the nonoscillation of the Sturm--Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm--Liouville type. [ABSTRACT FROM AUTHOR]

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions. [ABSTRACT FROM AUTHOR]

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This paper investigates the existence of Eberlein weakly almost periodic solutions for differential equations of the form u ′ = A u + f (t) and u ′ = A (t) u + f (t) . In the first scenario, when A generates a strongly asymptotically semigroup, we establish the existence of Eberlein-weakly almost periodic solutions, thereby extending and improving a previous result in Zaidman(Ann Univ Ferrara 14(1): 29–34, 1969). In the second case, we consider a more general situation where A(t) is a (possibly nonlinear) operator satisfying a monotony condition. Unlike most existing works in the literature, our approach does not rely on tools of exponential dichotomy and Lipschitz nonlinearity. Lastly, we illustrate the practical relevance of our findings by presenting real-world models, including a hematopoiesis model, that exemplify the key findings. A numerical simulation is also provided. [ABSTRACT FROM AUTHOR]

Investigating the role of chiral-induced spin selectivity in the generation of spin correlated radical pairs in a photoexcited donor–chiral bridge–acceptor system is fundamental to exploit it in quantum technologies. This requires a minimal master equation description of both charge separation and recombination through a chiral bridge. To achieve this without adding complexity and entering in the microscopic origin of the phenomenon, we investigate the implications of spin-polarizing reaction operators to the master equation. The explicit inclusion of coherent evolution yields non-trivial behaviors in the charge and spin dynamics of the system. Finally, we apply this master equation to a setup comprising a molecular qubit attached to the donor–bridge–acceptor molecule, enabling qubit initialization, control, and read-out. Promising results are found by simulating this sequence of operations assuming realistic parameters and achievable experimental conditions. [ABSTRACT FROM AUTHOR]

This paper introduces a hybrid spectral-conjugate gradient (SCG) method to solve nonlinear monotone operator equations efficiently. The proposed method incorporates a hybrid parameter that encompasses the Polak--Ribière--Polyak (PRP), Liu-Storey (LS), Fletcher-Reeves (FR), and conjugate descent (CD) methods as particular instances. Additionally, we derive the spectral parameter to ensure that the search direction adheres to the sufficient descent condition. The search direction is also designed to be bounded, and under specific conditions, we demonstrate that the sequence produced by our hybrid SCG algorithm converges toward a solution. Furthermore, to underscore the effectiveness of our proposed method, we conducted extensive numerical experiments comparing its performance against that of existing algorithms. These experiments were based on a selection of benchmark nonlinear monotone operator equations, highlighting our proposed algorithm's superior efficiency and potential in practical applications. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of dual periodic boundary conditions. One of these conditions relates to the endpoints of an interval, while the other one involves two nonlocal positions within the interval. Under these boundary conditions, we establish the criteria for the existence of solutions for a fractional differential equation with p-Laplacian operator by using Mawhin’s continuation theorem. An example is constructed for illustrating the obtained results. [ABSTRACT FROM AUTHOR]

The inverse scattering problem for the two-dimensional nonlinear Schrödinger equation i u t + Δ u = N (u) is studied. We assume that the unknown nonlinearity N of the equation satisfies N ∈ C ∞ (C ; C) , (∂ k ∂ ‾ ℓ N) (z) = O (| z | max ⁡ { 3 − k − ℓ , 0 }) (z → 0) and (∂ k ∂ ‾ ℓ N) (z) = O (e c | z | 2 ) (| z | → ∞) for any non-negative integers k and ℓ. Here, ∂ N and ∂ ‾ N are the Wirtinger derivatives of N , and c is a positive constant. We establish a reconstraction formula of (∂ k ∂ ‾ ℓ N) (0) (k + ℓ ≥ 3) by the knowledge of the scattering operator for the equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of determining the time‐dependent source term in the general setting of Hilbert spaces and for general additional data. We prove the well‐posedness of this inverse problem by reducing the problem to an operator equation for the source function. [ABSTRACT FROM AUTHOR]

In the present work, the transmutation operator approach is employed to construct an exact solution to the initial boundary-value problem for multidimensional free transverse equation vibration of a thin elastic plate with a singular Bessel operator acting on geometric variables. We emphasize that multidimensional Erdélyi–Kober operators of a fractional order have the property of a transmutation operator, allowing one to transform more complex multidimensional partial differential equations with singular coefficients acting over all variables into simpler ones. If th formulas for solutions are known for a simple equation, then we also obtain representations for solutions to the first complex partial differential equation with singular coefficients. In particular, it is successfully applied to the singular differential equations, particularly when they involve operators of the Bessel type. Applying this operator simplifies the problem at hand to a comparable problem, even in the absence of the Bessel operator. An exact solution to the original problem is constructed and analyzed based on the solution to the supplementary problem. [ABSTRACT FROM AUTHOR]

In this study, we aim to set new criteria regarding the asymptotic behavior of the neutral differential equation of the third order. These criteria are designed to ensure that this equation is oscillatory using comparisons with first-order differential equations and Riccati substitution. The results we obtained improve some of the results found in the literature. Some examples are provided to illustrate the applicability of our results and compare them with results found in some previous studies. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

We study a class of equations in Banach spaces with a Riemann–Liouville-type integrodifferential operator with an operator-valued convolution kernel. The properties of k-resolving operators of such equations are studied and the class Am,K,χ of linear closed operators is defined so that the belonging to this class is necessary and, in the case of commutation of the operator with the convolution kernel, sufficient for the existence of analytic in the sector k-resolving families of operators of the equation under study. Under certain additional conditions on the convolution kernel, we prove theorems on the unique solvability of the nonhomogeneous linear equation of the class under consideration if the nonhomogeneity is continuous in the norm of the graph of the operator from the equation or Hölder continuous. We obtain a theorem on sufficient conditions on an additive perturbation of an operator of the class Am,K,χ in order that the perturbed operator also belong to such a class. Abstract results are used in the study of initial-boundary value problems for a system of partial differential equations with several fractional Riemann–Liouville derivatives of different orders with respect to time and for an equation with a fractional Prabhakar derivative with respect to time. [ABSTRACT FROM AUTHOR]

Based on the extensive application of the q$$ q $$‐series and q$$ q $$‐polynomials including (for example) the q$$ q $$‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the q$$ q $$‐Laguerre polynomials. The mission of this paper is to find the general q$$ q $$‐operational equation together with the expansion issue of the bivariate q$$ q $$‐Laguerre polynomials from the perspective of q$$ q $$‐partial differential equations. We also give some applications including some q$$ q $$‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the U(n+1)$$ U\left(n&#x0002B;1\right) $$‐type generating functions for the bivariate q$$ q $$‐Laguerre polynomials by the technique based upon q$$ q $$‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate q$$ q $$‐Laguerre polynomials by applying q$$ q $$‐operational equations. U(n+1)$$ U\left(n&#x0002B;1\right) $$ [ABSTRACT FROM AUTHOR]

In a Hilbert space, we consider a second-order differential equation with unbounded operator coefficients modeling small motions of a dynamical system with a weak energy dissipation. A theorem on the existence and uniqueness of a classical solution is stated. The corresponding spectral problem is reduced to the study of an elliptic quadratic pencil, which, in turn, can be reduced to a "modified" Keldysh pencil. Depending on the asymptotics of the spectrum of the main operator of the problem (the potential energy operator) and the subordination coefficient of the energy dissipation operator, we prove that the corresponding root function system of the linearized problem is a double Bari basis, Riesz basis, or Abel–Lidskii basis with parentheses. As applications, we consider the problem on a quadratic Sturm–Liouville pencil as well as the general spectral problem generated by the problem on small motions of a system of hinged pendulums with cavities completely or partly filled with ideal incompressible fluids and with friction in hinges. [ABSTRACT FROM AUTHOR]

In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

A basic fact, having fundamental significance in quantum mechanics, is that hermitian (or self-adjoint) operators have only real eigenvalues. However, in certain applications in molecular physics, one deals with non-hermitian operators. We discuss a condition for non-hermitian operators to have real eigenvalues, proving that it is the case if and only if it can be decomposed as a product of two, generally non-commuting hermitian operators, one of which is positive definite. The theorem is illustrated on the example of non-hermitian effective Hamiltonians occurring in the non-perturbative form of the Bloch equation. [ABSTRACT FROM AUTHOR]