Bound Solutions Research Articles

Approximate bound state solutions of the fractional Schr\"{o}dinger equation under the spin-spin-dependent Cornell potential

In this work, the approximate bound state solutions of the fractional Schr\"{o}dinger equation under a spin-spin-dependent Cornell potential are obtained via the convectional Nikiforov-Uvarov approach. The energy spectra are applied to obtain the mass spectra of the heavy mesons such as bottomonium, charmonium and bottom-charm. The masses for the singlet and triplet spin numbers increase as the quantum numbers increase. The fractional Schr\"{o}dinger equation improves the mass spectra compared to other theoretic masses. The bottomonium masses agree with the experimental results where percentage errors for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were found to be 0.67\% and 0.49\% respectively. The respective percentage errors of 1.97\% and 1.62\% for fractional parameters of $\beta =1,\alpha =0.97$ and $\beta =1,\alpha =0.50$ were obtained for charmonium meson. The results indicate that the potential curves coupled with the fractional parameters account for the short-range gluon exchange between the quark-antiquark interactions and the linear confinement phenomena which is associated with the quantum chromo-dynamic and phenomenological potential models in particle and high-energy physics.

Analytical solution of fuzzy heat problem in two-dimensional case under Caputo-type fractional derivative.

This work aims to investigate the analytical solution of a two-dimensional fuzzy fractional-ordered heat equation that includes an external diffusion source factor. We develop the Sawi homotopy perturbation transform scheme (SHPTS) by merging the Sawi transform and the homotopy perturbation scheme. The fractional derivatives are examined in Caputo sense. The novelty and innovation of this study originate from the fact that this technique has never been tested for two-dimensional fuzzy fractional ordered heat problems. We presented two distinguished examples to validate our scheme, and the solutions are in fuzzy form. We also exhibit contour and surface plots for the lower and upper bound solutions of two-dimensional fuzzy fractional-ordered heat problems. The results show that this approach works quite well for resolving fuzzy fractional situations.

Limit analysis on roof failure mechanism of tunnels subjected to sequential excavations

Compared to traditional full-face excavation methods, sequential excavation methods (SEM), such as central diaphragm (CD) and sidewall drift (SD) methods, can significantly reduce rock disturbance and minimize the extent of roof failure in tunnel crowns. Therefore, they are particularly suitable for tunnel projects in weak rock formations and large cross-sectional underground chambers. This paper presents a kinematic approach to predict the roof failure mechanism of tunnels excavated using the CD method. The collapse blocks at the top are influenced by excavation steps and initial in-situ stresses, and the direction of their failure velocity field is typically non-vertical, posing challenges for limit analysis applications. To address this, a tilted velocity field for the collapse failure of the tunnel crown in the CD method is constructed based on the initial normal stress distribution along the tunnel’s top contour, obtaining an upper bound solution for tunnel collapse failure. The proposed method is compared with the full-face excavation method, demonstrating the superiority of the CD method over the full-section method. The rationality of the proposed tilted velocity field mechanism is verified through numerical validation using finite difference methods. Building upon the upper bound solution, safety assessment indicators for tunnels excavated using the CD method, namely allowable rock stress release and safety factor, are proposed. Furthermore, we extend this method to analyze the roof failure mechanism in tunnels constructed using the SD method and confirm its effectiveness through comparison with FDM analysis. Theoretical analysis and numerical validation both indicate that sequential excavation methods can significantly reduce the potential collapse range at the top and improve the safety of tunnel construction.

The calculation of ro-vibrational energies, franck-condon factors and r-centroids for Yttrium/Scandium oxides in D-dimension

The spectroscopic data as ro-vibrational energy values, Franck-Condon Factors and r-centroids are obtained for Yttrium/Scandium oxides with (B 2Σ+, X 2Σ+) different electronic states. Ro-vibrational energy values are calculated by bound state solution of the Schrödinger equation for the general molecular potential by applying Pekeris-type approximation to the centrifugal term in D-dimensional. Using these solutions, the relation giving the bound state energy eigenvalues and the corresponding wave functions expression are derived. The Franck-Condon factors and r-centroids values are determined by using Fraser-Jarmain method for the B 2Σ+ → X 2Σ+ electronic transitions in both molecules. All the results obtained are given numerically in the tables. The obtained values are compared with the previously get data in the literature and it is seen that the results are consistent.

Seismic active earth pressure for 3D earth retaining structure following a nonlinear failure criterion

Stability analysis for three-dimensional (3D) earth retaining structures (ERSs) is a pivotal problem in geotechnical engineering, particularly for seismically active areas. In this work, the 3D ERSs in soil are assessed for seismic stability following a nonlinear strength criterion. The power-law strength criterion and seismic forces are introduced into the 3D ERS stability analysis via a multi-cone failure mechanism. The coefficient of active soil pressure Ka is derived by exploiting the energy balance equation. The upper bound solutions, that is the maximum results of the Ka are captured with the help of a genetic algorithm. The validity of the current study is verified by a comparative analysis, and the effects of soil strength nonlinearity, the seismic forces, the soil-wall friction angle, the height, the inclined angle and the 3D geometric traits of ERSs on the Ka solutions and the stability of ERSs are investigated. It is indicated that the height and the 3D geometric characteristics of the ERSs will not only determine the Ka solutions directly, but also influence the impact of soil strength nonlinearity on the stability of ERSs. Strength criteria should be chosen combining the geometric characteristics of ERSs to derive more critical estimates on the stability of 3D ERSs. Additionally, a range of seismic and static stability charts used for preliminary design are put forward as well.

Modified pseudo-dynamic bearing capacity of reinforced soil foundations

In engineering practice, horizontal layers of reinforcement geosynthetics are widely used to improve the bearing capacity of foundations. However, there are few studies on the seismic bearing capacity of foundations on reinforced soil, and there is a lack of derivation of theoretical formulas in this direction. In this paper, based on the upper bound theorem of limit analysis, the seismic bearing capacity of foundations on reinforced soil is studied for the first time using the modified pseudo-dynamic approach. Seismic forces are calculated using a slicing method that adapts to spatial and temporal variations. Based on three different failure mechanisms, three different formulas and constraints are used. Each formula contains 61 variables, and the bearing capacity value is optimized by MATLAB program. By comparing the calculation results of three failure mechanisms, the exact upper bound solution of seismic bearing capacity for sliding failure is obtained. Considering the temporal as well as spatial characteristics of seismic forces, the effects of the coefficient of seismic acceleration, the internal friction angle, the damping ratio, the normalized frequency, and the reinforcement burial depth on the seismic bearing capacity are analyzed, providing a theoretical basis for the application of reinforced soil foundations in engineering practice.

Solutions of Schrödinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP) within the frame of parametric Nikiforov-Uvarov method

In this paper, we deduced an approximate bound state solutions of Schrӧdinger equation based on Modified Exponential Screened Plus Yukawa Potential (MESPYP), with the help Greene–Aldrich Approximation to determine the centrifugal term. The analytical solution was obtained based on Nikiforov–Uvarov (NU) method. The bound state solutions of five diatomic molecules which are mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also calculated numerically. The estimated energy eigenvalues for these molecules were deduced using the resulting energy eigenequation and total unnormalized wavefunction is expressed in term of Jacobi polynomial. The numerical solution shows that the bound state solutions increase with increase in the quantum state.

Topological defects on the energy spectra of diatomic molecules embedded with shifted Deng–Fan oscillator potential under AB flux field

AbstractA study on the effect of point like global monopole topological defects on the energy eigenvalues of the diatomic molecules , , , embedded with Shifted Deng–Fan Oscillator Potential under the influence of Aharonov–Bohm flux field has been made here. Asymptotic Iteration Method (AIM) is used to find out the bound state solutions for arbitrary states by solving the non‐relativistic Schrödinger equation. A Pekeris‐type approximation has been used to approximate the centrifugal barrier term. It is observed that, energy levels of the diatomic molecules is significantly affected by the global effects of the point like global monopole, flux field and effective potential field.

Existence of a positive bound state solution for logarithmic Schrödinger equation

We investigate the existence of bound state solutions for the logarithmic Schrödinger equation−Δu+V(x)u=ulog⁡u2inRN,N≥1, where V(x)∈C(RN) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.

Exact and approximate bound state solutions of the Schrödinger equation with a class of Kratzer-type potentials in the global monopole spacetime

This work investigates the motion of a non-relativistic charged particle in the spacetime of a global monopole. We introduce the Schrödinger equation to describe the particle’s motion with two interactions by considering the Kratzer and the screened modified Kratzer potential. The problem’s eigenfunctions and eigenvalues are obtained by deriving and solving the radial equation. The effective potential encompasses both the Kratzer and electrostatic self-interaction potential and leads to bound states solutions. The energy spectrum is investigated, particularly emphasizing its dependence on the system’s physical parameters. The screened modified Kratzer potential, and the screened self-interaction potential reveal an important role in influencing both the effective potential and the energy spectrum. Additionally, it also accommodates the existence of bound states. All these behaviors are illustrated with graphs and discussed in detail.

Energy eigenvalues and finite-temperature magnetization for the improved Scarf II potential in the presence of external magnetic and Aharonov-Bohm flux fields

In this paper, the bound state solutions of the radial Schrödinger equation are obtained in closed form under an improved Scarf II potential energy function (ISPEF) constrained by external magnetic and Aharonov-Bohm (AB) flux fields. By constructing a suitable Pekeris-like approximation scheme for the centrifugal barrier, approximate analytical expressions for the bound-states and thermal partition function were obtained. With the aid of the partition function, an explicit equation for magnetization at finite temperatures is developed. The obtained equations were then applied to calculate the energy levels and magnetic properties of 7Li2 (2 3Πg), K2 (X 1Σg+), Mg2 (X 1Σg+) and NaBr (X 1Σ+) diatomic molecules. The obtained numerical results of the vibrational energies for these molecules were found to be in good agreement with theoretic and experimental values reported in the existing literature. The results indicated that by turning off the magnetic and AB fields, the energy levels of the diatomic molecules degenerate. The results further revealed that an increase in the temperature of the molecules and the AB field strengths leads to a linear decrease in magnetization.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

Estimation Criteria for Static Rock Mass Deformability Modulus for Rock-Socket Design in Metamorphic Rock Masses

This study investigates the most appropriate empirical criteria to estimate the static rock mass deformability modulus (𝐸𝑚) in the design of rock-sockets in cast-in-situ bored pile construction. The in-situ 𝐸𝑚 values are initially estimated through back analysis of static pile load test data. Secondly, the rock mass deformability estimated from back analysis (𝐸𝑚b) are tested statistically against selected established empirical equations to determine whether the latter are appropriate for use in metamorphic rock terrain of Sri Lanka. It is found that the existing empirical criterion based on the square root of intact unconfined compressive strength (𝜎𝑐) derived from back analysis of pile load test results is appropriate for weak-poor rock masses. For strong-poor rocks, it is recommended to employ the equation based on 𝜎𝑐', and in general the two equations generate lower and upper bound solutions. The equation based on intact deformability modulus (𝐸𝑖) performs well in strong-excellent quality rock masses, while the equations based on 𝐸𝑖 and rock quality designation (𝑅QD) are found to be appropriate for weak-fair to excellent rock masses. Finally, a new set of equations appropriate for different rock mass types have been proposed through regression analysis along with appropriate design measures to be adopted.

Klein–Gordon oscillator under gravitational effects in a topologically charged Ellis–Bronnikov wormhole

In this paper, we investigate the phenomenon of a relativistic quantum oscillator in non-massive wormhole spacetime, known in the literature as Ellis–Bronnikov-type wormhole, for a spin-zero particle subjected to the effects of a scalar field. We have obtained analytically the bound state solutions and your respective energy spectrum. The energy profile of this scalar field is modified by the wormhole throat as well as the scalar coupling constant of curvature. In the second part of the work, we investigate how the energy spectrum for a spin-zero particle is modified by an Aharonov–Bohm geometric phase effect. We also built a graphical representation to try to visualize this modification generated by the magnetic flux constant.

Information theory and thermodynamic properties of diatomic molecules using molecular potential.

Owing to the devise applications of molecules in industries, the bound state solution of the non-relativistic wave equation with a molecular potential function has been obtained in a closed-form using the Nikiforov-Uvarov method. The solutions of the bound state are then applied to study the information-theoretic measures such as the one-dimensional Shannon and Renyi entropic densities. The expectation values for the position and momentum spaces were obtained to verify the Heisenberg's uncertainty principle. Utilizing the energy spectrum equation, the thermodynamic vibrational partition function is obtained via the Poisson summation. Other thermodynamic function variations with absolute temperature have been obtained numerically for four diatomic molecules (H2, N2, O2, and HF) using Maple 18 software. The Shannon global entropic sum inequality has also been verified. The Renyi sum for constrained index parameters satisfies the global entropic inequality. The thermodynamic properties of the four molecules are similar and conform to works reported in the existing literature. The obtained vibrational energies are in fair agreement with the ones obtained using other forms of potential energy. The result further indicates that the lowest bounds for the Shannon, Renyi, and Heisenberg inequalities are ground states phenomena.

Approximate Analytical Solution of Fuzzy Linear Volterra Integral Equation via Elzaki ADM

In this paper, the fuzzy Volterra integral equations’ solutions are calculated using a hybrid methodology. The combination of the Elzaki transform and Adomian decomposition method results in the development of a novel regime. The precise fuzzy solutions are determined using Elzaki ADM after the fuzzy linear Volterra integral equations are first translated into two crisp integral equations utilizing the fuzzy number in parametric form. Three instances of the considered equations are solved to show the established scheme’s dependability, efficacy, and application. The results have a substantial impact on the fuzzy analytical dynamic equation theory. The comparison of the data in a graphical and tabular format demonstrates the robustness of the defined regime. The lower and upper bound solutions’ theoretical convergence and error estimates are highlighted in this paper. A tolerable order of absolute error is also obtained for this inquiry, and the consistency of the outcomes that are approximated and accurate is examined. The regime generated effective and reliable results. The current regime effectively lowers the computational cost, and a faster convergence of the series solution to the exact answer is signaled.

Multi-period facility location and capacity expansion with modular capacities and convex short-term costs

In this paper, we consider a multi-period facility location problem with capacity expansion motivated by the real-world problem of establishing hydrogen production infrastructure in Norway. The problem is formulated using modular capacities that capture economies of scale in production costs. The costs of opening a facility are represented by concave long-term costs, while the production costs of each capacity level are given by convex short-term costs. In our model, we allow only one expansion during the planning horizon, and have to observe limits on minimum production quantities. The objective is to minimize the sum of investment, expansion, production, and distribution costs while satisfying customer demand. To solve the problem we implement a solution method based on Lagrangian relaxation. The lower bound is calculated using a dynamic programming approach. To obtain an upper bound solution, we develop a greedy heuristic that converts the solution to the Lagrangian dual into a feasible solution. The approach is tested on different problem instances based on real-world data. The results show that our solution method based on Lagrangian relaxation outperforms Gurobi in terms of run time for all tested instances. Our Lagrangian based approach also always finds good or even near-optimal solutions, whereas Gurobi fails to find feasible solutions for some of the larger instances.

An Analytical Framework for Assessing the Unsaturated Bearing Capacity of Strip Footings under Transient Infiltration

The evaluation of the bearing capacity of strip footings generally assumes that the soil is either dry or fully saturated, which contradicts the actual condition in nature where the soil is often in a partially saturated state. Furthermore, infiltration has a significant impact on the shear strength of the soil. Following the upper bound theory of the limit analysis, this article provides a theoretical framework for assessing the bearing capacity under transient flow with linear variation in infiltration intensity for the first time. Firstly, the closed form of suction stress under linear transient infiltration is derived using Laplace transform and introduced into the Mohr–Coulomb criterion. A discrete failure mechanism with fewer variables and higher accuracy is provided to ensure kinematic admissibility. The upper bound solution for bearing capacity is obtained by solving the power balance equation. The present results are compared with results from the published literature and the finite element, confirming the validity and superiority of the theoretical framework provided. A parametric analysis is also conducted on three hypothetical soil types (fine sand, silt, and clay), and the results show that unsaturated transient infiltration has a positive influence on increasing the foundation bearing capacity. The magnitude of the influence is comprehensively controlled by factors such as soil type, saturated hydraulic conductivity, infiltration intensity, infiltration time, and water table depth. The increase in bearing capacity due to unsaturated transient infiltration can be incorporated into Terzaghi’s equation as a separate component presented in tabular form for engineering design purposes.

Bounded Solutions of Semi-Linear Parabolic Differential Equations with Unbounded Delay Terms

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is established. The application of the main theorem to four different semi-linear delay parabolic differential equations is presented. The first- and second-order accuracy difference schemes (FSADSs) for the solution of a one-dimensional semi-linear time-delay parabolic equation are considered. The new desired numerical results of this paper and their discussion are presented.

Vandermonde Operator and Bounded Solutions of Difference Equations

Vandermonde Operator and Bounded Solutions of Difference Equations