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This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory. [ABSTRACT FROM AUTHOR]

Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler's fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution's existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example. [ABSTRACT FROM AUTHOR]

In this paper, we considered the existence of solutions to a discrete second-order resonance problem with eigenparameter-dependent boundary conditions. We first transformed the resonance problem into its corresponding equivalent system using the Lyapunov-Schmidt method. In addition, using Schauder's fixed-point theorem and the connectivity theories of the solution set of compact vector fields, we obtained the existence and multiplicity of solutions to the second-order discrete resonance problem with eigenparameter-dependent boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we studied a Hadamard-type fractional Riemann-Stieltjes integral boundary value problem. The existence of nontrivial solutions was obtained by using the fixed-point method when the nonlinearities can be superlinear, suberlinear, and have asymptotic linear growth. Our results improved and generalized some results of the existing literature. [ABSTRACT FROM AUTHOR]

Let α ∈ (1 , 2 ] , β ∈ (0 , 1) with α − β > 1. This paper focused on the multiplicity of positive solutions for a singular tempered fractional boundary value problem { − 0 R D t α , λ u (t) = p (t) h (e λ t u (t) , 0 R D t β , λ u (t)) , t ∈ (0 , 1) , 0 R D t β , λ u (0) = 0 , 0 R D t β , λ u (1) = 0 , where h ∈ C ([ 0 , + ∞) × [ 0 , + ∞) , [ 0 , + ∞)) and p ∈ L 1 ([ 0 , 1 ] , (0 , + ∞)). By applying reducing order technique and fixed point theorem, some new results of existence of the multiple positive solutions for the above equation were established. The interesting points were that the nonlinearity contained the lower order tempered fractional derivative and that the weight function can have infinite many singular points in [ 0 , 1 ]. [ABSTRACT FROM AUTHOR]

In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking ε -approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of u (m) (x) . In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields extremely accurate approximation to the exact solution. Meanwhile, compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective and convenient. [ABSTRACT FROM AUTHOR]

In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature. [ABSTRACT FROM AUTHOR]

Probabilistic domain decomposition (PDD) is an alternative paradigm for solving boundary value problems (BVPs) in parallel with excellent scalability properties, thanks to its reliance on stochastic representations of the BVP. However, there are cases when the latter is less numerically convenient, or unknown. Semilinear elliptic BVPs and the Helmholtz equation are prominent examples of either class. In this paper, we overcome this issue by designing suitable iterative schemes for either problem. These schemes not only retain the desirable properties of PDD but also are optimally suited for pathwise variance reduction, resulting in a systematic, nearly cost-free reduction of the statistical error through the iterations. Numerical tests carried out on the supercomputer Marconi100 are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed boundary value problem to a class of nonlinear operators containing p(·)-Laplacian. More precisely, we are concerned with the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least three weak solutions under some hypotheses on given functions and the values of parameters. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions of the following third-order three-point boundary value problem with p-Laplacian {(Φp(u"(t)))' + f(t, u(t)) = 0, t (0, 1), u(0) = au(η), u(1) = au(η), u"(0) = 0, is studied, where Φp(s) = |s|p-2s, p > 1. By using the fixed point index method, we establish sufficient conditions for the existence of at least one or at least two positive solutions for the above boundary value problem. The main result is demonstrated by providing an example as an application. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities. [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's 13rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce the fourth and sixth-order compact finite difference methods for solving anti-periodic boundary value problems. Compact finite difference formulas can approximate the derivatives of a function more accurately than the standard finite difference formulas for the same number of grid points. The convergence analysis of the proposed method is also investigated. This analysis shows how the error between the approximate and exact solutions decreases as the grid space is reduced. To validate the proposed method's accuracy and efficiency, some computational experiments are provided. Moreover, a comparison is performed between the standard and compact finite difference methods. The experiments indicate that the compact finite difference method is more accurate and efficient than the standard one. [ABSTRACT FROM AUTHOR]

Monge–Ampère equations have important research significance in many fields such as geometry, convex geometry and mathematical physics. In this paper, under some superlinear and sublinear conditions, the existence of nontrivial solutions for a system arising from Monge–Ampère equations with two parameters is investigated based on the Guo–Krasnosel'skii fixed point theorem. In the end, two examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φ p -Laplacian for the discrete 2 n -th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied. [ABSTRACT FROM AUTHOR]

This paper demonstrates the utilization of Pontryagin Neural Networks (PoNNs) to acquire control strategies for achieving fuel-optimal trajectories. PoNNs, a subtype of Physics-Informed Neural Networks (PINNs), are tailored for solving optimal control problems through indirect methods. Specifically, PoNNs learn to solve the Two-Point Boundary Value Problem derived from the application of the Pontryagin Minimum Principle to the problem's Hamiltonian. Within PoNNs, the Extreme Theory of Functional Connections (X-TFC) is leveraged to approximate states and costates using constrained expressions (CEs). These CEs comprise a free function, modeled by a shallow neural network trained via Extreme Learning Machine, and a functional component that consistently satisfies boundary conditions analytically. Addressing discontinuous control, a smoothing technique is employed, substituting the sign function with a hyperbolic tangent function and implementing a continuation procedure on the smoothing parameter. The proposed methodology is applied to scenarios involving fuel-optimal Earth−Mars interplanetary transfers and Mars landing trajectories. Remarkably, PoNNs exhibit convergence to solutions even with randomly initialized parameters, determining the number and timing of control switches without prior information. Additionally, an analytical approximation of the solution allows for optimal control computation at unencountered points during training. Comparative analysis reveals the efficacy of the proposed approach, which rivals state-of-the-art methods such as the shooting technique and the adaptive Gaussian quadrature collocation method. [ABSTRACT FROM AUTHOR]

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In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of multiple positive solutions for a class of φ-Laplacian boundary value problem with a nonlinear fractional differential equation and fractional boundary conditions. Multiple solutions are proved under slight conditions on a possibly degenerating source term. Approximation techniques together with the fixed point index theory a on cone of a Banach space are employed. Some illustrating examples of are also supplied. [ABSTRACT FROM AUTHOR]

In this paper, we compute holographic Euclidean thermal correlators of the stress tensor and U(1) current from the AdS planar black hole. To this end, we set up perturbative boundary value problems for Einstein's gravity and Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with appropriate gauge fixing and regularity boundary conditions at the horizon of the black hole. The linearized Einstein equation and Maxwell equation in the black hole background are related to the Heun equation of degenerate local monodromy. Leveraging the connection relation of local solutions of the Heun equation, we partly solve the boundary value problem and obtain exact two-point thermal correlators for U(1) current and stress tensor in the scalar and shear channels. [ABSTRACT FROM AUTHOR]

Temperature evolution during plastic deformation is of great importance for the design of manufacturing processes, as well as for the analysis and prediction of tool wear. However, the results from experimental- and numerical-type research are still often contradictory. In this paper, we analyze methods for estimating plasticity-induced heating directly from displacement fields that can be recorded during experiments or extracted from simulation results. In terms of computational methodology, the thermodynamically motivated energy-based variational formulation of the coupled thermo-mechanical boundary-value problem is adapted to the problem at hand. Since an analysis of this variational formulation exhibits challenges and distinct inconsistencies with respect to the problem at hand, an alternative approach is proposed. This alternative approach is essentially a purely thermal finite element simulation, and it is conducted using a heat source term that is empirically based on the fraction of irreversible deformation work converted to heat. Our approach estimates plasticity-induced heating based on the strain and strain rate data derived from displacement fields. We therefore incorporate thermo-visco-plastic constitutive behavior (Johnson–Cook) with a thermodynamically motivated model that specifies the fraction of plastic work converted to heat (the Taylor–Quinney coefficient). [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to define a new additive function related to generalized divisors of an integer and to study some of its properties. [ABSTRACT FROM AUTHOR]

In this paper, we present a continuous model as an analog to the discrete atomic model often used in the interpretation/analysis of magnetometry studies of magnetic structures with bilinear and/or biquadratic coupling. For a typical set of parameters, the continuous model is shown to be nearly indistinguishable from its discrete counterpart; validating it for data analysis purposes. Furthermore, we also find a very simple analytic expression which can approximately solve the continuous model and also provides insight into the behaviors of the magnetic moment within the magnetic structure. The main advantage of the continuous model is that numerically it can be solved as a boundary value problem, which can be significantly faster than the energy minimization required for the discrete model improving computational speed and allowing for much more rapid data analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters. [ABSTRACT FROM AUTHOR]

SUMMARY: When the conventional finite-element method is used to simulate the 3-D direct-current (DC) resistivity response over a conductive earth with large complex structures, it requires finely discretized mesh to accurately represent the underground structures. Directly solving the current-conduction problem on the fine mesh will lead to a huge amount of calculation. In this paper, we develop a fast 3-D forward modelling method for DC resistivity method based on smooth multiscale finite-element algorithm. Instead of using the conventional polynomial basis functions, we construct the multiscale basis functions by solving the local boundary-value problems of partial differential equation in parallel on the multiscale meshes. The multiscale basis functions can capture the small-scale heterogeneous information in coarse cells and reflect it to the large scale by assembling macro matrix of coarse mesh. Thus, it enables us to quickly obtain the accurate solution by solving the original current-conduction problem with complex structures on coarse mesh. We further adopt the oversampling technology to improve the forward modelling accuracy. Besides, by combining with the gradient smoothing technology, we avoid establishing the continuous form of multiscale basis functions and their spatial derivative integral operation to rapidly assemble the macro matrix. Finally, the reliability of the proposed algorithm is verified by applying our code to the 3-D complex models and comparing it with the conventional finite-element method. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and uniqueness of local regular solutions to the initial-Neumann boundary value problem of the Schrödinger flow from a smooth bounded domain Ω ⊂ R 3 into S 2 (namely Landau–Lifshitz equation without dissipation). The proof is built on a parabolic perturbation method, an intrinsic geometric energy argument, the symmetric (algebraic) properties of S 2 and some observations on the behaviors of some geometric quantities on the boundary of the domain manifold.It is based on methods from Ding and Wang (one of the authors of this paper) for the Schrödinger flows of maps from a closed Riemannian manifold into a Kähler manifold as well as on methods by Carbou and Jizzini for solutions of the Landau–Lifshitz equation. [ABSTRACT FROM AUTHOR]

In this paper, we apply the method of asymptotic homogenization in parametric space in order to determine thermo-viscoelastic properties of materials composed of cylindrical fibers surrounding by an intermediate layer separating them from viscoelastic matrix. We use the Maxwell model of viscoelasticity. The volumetric fraction and physical properties of the layer can vary. There is a possibility of a contrast between the properties of fibers, the layer, and the matrix. Our approach allows us to treat temperature as a parameter and study the response of mechanical characteristics on the change in temperature. We perform the Fourier transform of governing equations and then implement asymptotic homogenization of the resulting equations in parametric space. Microscopic boundary value problem is defined on a periodic cell consisting of a fiber surrounded by a layer and embedded in the matrix. Macroscopic equations are derived for the complex Young's and shear moduli. Effective viscoelastic properties of the material are computed for different values of the frequency of harmonic oscillations and for various temperatures. This paper discusses how effective storage and loss Young's and shear moduli depend on volumetric fraction and mechanical properties of the material of the layer. The obtained results can provide information on how concentration, structure and properties of inclusions can affect the effective storage and loss Young's and shear moduli. [ABSTRACT FROM AUTHOR]

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω , ρ) -BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer's fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions for a singular tempered fractional equation with a p-Laplacian operator. By constructing a pair of suitable upper and lower solutions of the problem, some new results on the existence of positive solutions for the equation including singular and nonsingular cases are established. The asymptotic behavior of the solution is also derived, which falls in between two known curves. The interesting points of this paper are that the nonlinearity of the equation may be singular in time and space variables and the corresponding operator can have a singular kernel. [ABSTRACT FROM AUTHOR]

This paper aims to develop an effective theoretical method for analyzing the interaction of interfacial crack and complex defect in functionally graded magnetic-electro-elastic bi-materials with exponential variation under the action of anti-plane incident SH-wave. The boundary value problem of interest is solved by Green's function method. The mechanical model of the cracks is constructed through interface conjunction and crack deviation techniques, thereby simplifying the crack problem to solving a series of the first kind of Fredholm's integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tips of the left crack are expressed. The validity of the present method is verified by comparing with a single crack model in previous work. Through the analyses of numerical cases, it can be concluded that the DSIF is related with 3 factors：the geometry of complex defect and crack, the characteristics of incident wave and the inhomogeneity of materials. Comparing with dual integral equations in dealing with asymmetric defects, the method in this paper is relatively more flexible and applicable, also it comes up with a new way to study fracture problems in functionally graded magnetic-electro-elastic materials with more complex defects. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of solutions for a general class of second-order nonlinear impulsive boundary value problems defined on an infinite interval. The main innovative aspects of the study are that the results are obtained under relatively mild conditions and the use of principal and nonprincipal solutions that were obtained in a very recent study. Additional results about the existence of bounded solutions are also provided, and theoretical results are supported by an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, the Neumann boundary value problem for the Beltrami operator is explicitly solved in a circular ring domain, solvability conditions for this problem are also given in explicit forms. Moreover, the Neumann problem for second-order operators with the Bitsadze/Laplace operator as the main part as combinations of the Cauchy-Riemann and the Beltrami operators is investigated. [ABSTRACT FROM AUTHOR]

In this paper, using the notion of ( I q )–stability and the method of a priori estimates, known as the method of lower and upper solutions, the sufficient conditions guaranteeing uniform convergence of solutions to the solution of a reduced problem on the entire interval [ a , b ] have been established for four different types of boundary conditions for a singularly perturbed differential equation ε y ″ = f (x , y , y ′) ,   a ≤ x ≤ b . In the second part of the paper, by employing the Peano phenomenon, we analyzed the structure of the solutions of the reduced problem f (x , y , y ′) = 0 . [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a chemotaxis system in a two-dimensional setting as follows: { u t = Δ u − χ ∇ ⋅ (u ∇ ln ⁡ v) − κ u v + r u − μ u 2 + h 1 , v t = Δ v − v + u v + h 2 , with the parameters χ , κ , μ > 0 and r ∈ R , and with the given functions h 1 , h 2 ≥ 0. This model was originally introduced by Short et al for urban crime with the particular values χ = 2 , r = 0 and μ = 0 , and the logistic source term r u − μ u 2 was incorporated into ( ⋆) by Heihoff to describe the fierce competition among criminals. Heihoff also proved that the initial-boundary value problem of ( ⋆) possesses a global generalized solution in the two-dimensional setting. The main purpose of this paper is to show that such a generalized solution becomes bounded and smooth at least eventually. In addition, the long-time asymptotic behavior of such a solution is discussed. [ABSTRACT FROM AUTHOR]

This article concerns the existence of positive weak solutions of a heterogeneous elliptic boundary value problem of logistic type in a very general annulus. The novelty of this work lies in considering non-classical mixed glued boundary conditions. Namely, Dirichlet boundary conditions on a component of the boundary, and glued Dirichlet-Neumann boundary conditions on the other component of the boundary. In this paper we perform a complete analysis of the existence of positive weak solutions of the problem, giving a necessary condition on the parameter for the existence of them, and a sufficient condition for the existence of them, depending on the -parameter, the spatial dimension and the exponent of the reaction term. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results obtained in this paper are pioners in the field, because up the knowledge of the autor, this is the first time where this kind of logistic problems have been analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider an iterative system of Hadamard fractional boundary value problems. The sufficient conditions are derived for the existence of at least one positive solution by utilizing Guo-Krasnosel'skii fixed point theorem. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial boundary value problem for the one-dimensional thermodiffusion equations of type III. By the semigroup approach and the energy method, we establish the global existence and exponential stability for the solutions in a bounded region. The main work of this paper is to extend the study of thermodiffusion equations to the model of type III, which can be used as a reference for the study of other types of partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the functional differential equation (D C a + α x) (t) + ∑ i = 0 m (T i x (i)) (t) = f (t) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green's functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green's functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (D C a + α ν) (t) + ∑ i = 0 m (T i v (i)) (t) < 0 , t ∈ [ a , b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green's functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for "small" deviations. [ABSTRACT FROM AUTHOR]

The notion of competing HT ht -Laplacians with weights is considered for the first time. They are driven by a degenerated HT ht -Laplacian with weights and a competing HT ht -Laplacian with weights, respectively. Finally, the authors indicate how the result can be extended to HT ht -equations ( HT ht ). This research aims to present a linear operator HT ht by utilizing the I q i -Mittag-Leffler function and to introduce the subclass of harmonic HT ht -convex functions HT ht related to the Janowski function. [Extracted from the article]

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in ψ -fractional space H p α , β ; ψ (Ω) . The results obtained in this paper are the first to make use of the theory of ψ -Hilfer fractional operators with p-Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, we solve a linear system of second-order boundary value problems by using the quadratic B-spline finite element method (FEM). The performance of the method is tested on one model problem. Comparisons are made with both the analytical solution and some recent results.The obtained numerical results show that the method is efficient. [ABSTRACT FROM AUTHOR]

In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition, Ψ=Ψ1+Ψ2, was presented for stationary bound states Ψ of the one-dimensional Schrödinger equation, such that the components Ψ1 and Ψ2 approach their semiclassical WKB analogs in the large action limit. Moreover, by applying the Madelung-Bohm ansatz to the components rather than to Ψ itself, the resultant bipolar Bohmian mechanical formulation satisfies the correspondence principle. As a result, the bipolar quantum trajectories are classical-like and well behaved, even when Ψ has many nodes or is wildly oscillatory. In this paper, the previous decomposition scheme is modified in order to achieve the same desirable properties for stationary scattering states. Discontinuous potential systems are considered (hard wall, step potential, and square barrier/well), for which the bipolar quantum potential is found to be zero everywhere, except at the discontinuities. This approach leads to an exact numerical method for computing stationary scattering states of any desired boundary conditions, and reflection and transmission probabilities. The continuous potential case will be considered in a companion paper [C. Trahan and B. Poirier, J. Chem. Phys. 124, 034116 (2006), following paper]. [ABSTRACT FROM AUTHOR]