Laplace Transform Approach Research Articles

Relativistic effect of pseudospin symmetry and tensor coupling on the Mie-type potential via Laplace transformation method

A relativistic Mie-type potential for spin-1/2 particles is studied. The Dirac Hamiltonian contains a scalar S(r) and a vector V(r) Mie-type potential in the radial coordinates, as well as a tensor potential U(r) in the form of Coulomb potential. In the pseudospin (p-spin) symmetry setting Σ = Cps and Δ = V(r), an analytical solution for exact bound states of the corresponding Dirac equation is found. The eigenenergies and normalized wave functions are presented and particular cases are discussed with any arbitrary spin—orbit coupling number κ. Special attention is devoted to the case Σ = 0 for which p-spin symmetry is exact. The Laplace transform approach (LTA) is used in our calculations. Some numerical results are obtained and compared with those of other methods.

Laplace Transformation Approach to the Spin Symmetry of the Mie-Type Potential with a Coulomb Tensor Interaction

The Dirac equation is solved exactly under the condition of spin symmetry for a spin 1=2 particle in the field of Mie-type potential and a Coulomb-like tensor interaction via the Laplace transform approach (LTA). The Dirac bound state energy equation and the corresponding normalized wave functions are obtained in closed forms with any spin-orbit coupling term k. The effects of the tensor interaction and the potential parameters on the bound states are also studied. It is noticed that the tensor interaction removes degeneracy between two states in spin doublets. Some numerical results are given and a few special cases of interest are presented. We have demonstrated that in the nonrelativistic limit, the solutions of the Dirac system converges to that of the Schrödinger system. The nonrelativistic limits of the present solutions are compared with the ones obtained by findings of other methods. Our results are sufficiently accurate for practical purpose.

New transient-flow modelling of a multiple-fractured horizontal well

A new transient-flow modelling of a multiple-fractured horizontal well is presented. Compared to conventional modelling, the new modelling considered more practical physical conditions, such as various inclined angles for different fractures, different fracture intervals, different fracture lengths and partially penetrating fractures to formation. A kind of new mathematical method, including a three-dimensional eigenvalue and orthogonal transform, was created to deduce the exact analytical solutions of pressure transients for constant-rate production in real space. In order to consider a wellbore storage coefficient and skin factor, we used a Laplace-transform approach to convert the exact analytical solutions to the solutions in Laplace space. Then the numerical solutions of pressure transients in real space were gained using a Stehfest numerical inversion. Standard type curves were plotted to describe the transient-flow characteristics. Flow regimes were clearly identified from type curves. Furthermore, the differences between the new modelling and the conventional modelling in pressure transients were especially compared and discussed. Finally, an example application to show the accordance of the new modelling with real conditions was implemented. Our new modelling is different from, but more practical than, conventional modelling.

A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models

Solvable models of the Schrödinger equation are important models of quantum systems because they are idealistic approximations of real quantum systems and much insight into real quantum systems can be gained from the exact solutions of the solvable models. In this paper we show that a universal Laplace transform scheme can be used to solve the Schrödinger equations in closed form for all known solvable models. The work demonstrates how to apply the Laplace transform to differential equations with non-constant coefficients, which is useful in many branches of physics in addition to quantum mechanics. The advantages of the Laplace transform over the power expansion method and its connection with the methods of supersymmetry shape-invariant potentials and quantum canonical transformation, which also give closed-form solutions for solvable models, are elucidated.

Makarov potential in relativistic equation via Laplace transformation approach

Exact solutions and corresponding normalized eigenfunctions of the Dirac equation are studied for the Makarov potential by using the Laplace transform approach under the pseudospin symmetry. By using the ideas of SUSY and shape invariance, we obtain the energy eigenvalues equation. The wave functions of the angle part are obtained by using the Nikiforov–Uvarov method, too. Finally, we also discuss the special cases of this potential and the valence energy states can be produced from our solution for the hole state by taking appropriate transformation of parameters.

Exact Solutions of the Morse-like Potential, Step-Up and Step-Down Operators via Laplace Transform Approach

We intend to realize the step-up and step-down operators of the potential V(x) = V1e2βx + V2eβx. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential (β → −β).

Linear analysis of a rectangular waveguide cyclotron maser with a sheet electron beam

A linear theory for a rectangular waveguide cyclotron maser with a sheet electron beam is developed by using the Laplace transformation approach. This theory can be applied to any TEmn rectangular waveguide mode. The corresponding equations for the TMmn mode in the rectangular waveguide are also derived as a useful reference. Especially, the effect from the coupling between degenerate modes, which is induced by the nonideal rectangular waveguide walls, on the dispersion relation is considered in order to provide a more accurate model for the real devices. Through numerical calculations, the linear growth rate, launching loss, and spontaneous oscillations (caused by the absolute instability and backward wave oscillation) of this new structure can be analyzed in detail. It is worthwhile to point out that the operation at higher power levels of the rectangular waveguide sheet beam system is possible.

Brownian excursions and Parisian barrier options: a note

This paper addresses Paris barrier options, as introduced by G. Kentwell and J. Cornwall at Bankers Trust Australia in the mid-1990s, and their valuation, as developed by Chesnay, Jeanblanc-Picqué and Yor using the Laplace-transform approach. The notion of Paris barrier options is extended so that their valuation becomes possible at any point during their lifespan, and the pertinent Laplace transforms of Chesnay, Jeanblanc-Picqué and Yor are modified when necessary.

Brownian excursions and Parisian barrier options: a note

This paper addresses Paris barrier options, as introduced by G. Kentwell and J. Cornwall at Bankers Trust Australia in the mid-1990s, and their valuation, as developed by Chesnay, Jeanblanc-Picqué and Yor using the Laplace-transform approach. The notion of Paris barrier options is extended so that their valuation becomes possible at any point during their lifespan, and the pertinent Laplace transforms of Chesnay, Jeanblanc-Picqué and Yor are modified when necessary.

Erratum to: Light propagation in weakly guiding optical fibres. A Laplace-transform approach

Erratum to: Light propagation in weakly guiding optical fibres. A Laplace-transform approach

Light propagation in weakly guiding optical fibres. A Laplacetransform approach

In this paper we study the scalar Helmholtz equation, in cylindrical coordinates, for the propagation of light in a weakly guiding optical fibre with the refraction index sechR (R is the distance from the axis of the fibre). We accomplish this study with the help of the Laplace-transform method which proves to be particularly well suited to the purpose. The analysis around the singular points of the Laplace-transformed equation provides not only the asymptotic behaviour of the solution of the original equation (which is a general feature of the Laplace transformation) but can even be transferred to the region of regularity, more precisely around a point which is singular for the equation but regular for the solution. In principle, this makes the search of the eigenvalue in the Laplacetransformed space easier because the matching of the asymptotic solution with the solution in the left half-plane (in turn equivalent, in the original space, to the matching of the regular solution on the right of the origin with the asymptotic vanishing solution) is also «transferred» in the right half-plane, which is a regularity region in the above-mentioned sense. Cut-off frequencies (in particular for the second mode) arise at the same manner: a simple and general relation is established for them. We stop the analysis at the stage in which the asymptotic expansion of the Laplace transform around the point at infinity is still to be conveniently represented («summed»). However, the crude approximation of using only the first term of the expansion yields already encouraging results.

A Combined Laplace Transform and Streamline Upwind Approach for Nonideal Transport of Solutes in Porous Media

A new numerical method for nonideal transport of solutes in porous media is developed by first using the Laplace transform to eliminate the time dependency and then the streamline upwind scheme to solve the spatial equations. The transient solution is ultimately recovered by an efficient quotient difference inversion algorithm. By introducing complex‐valued artificial dispersion in the weighting functions, characteristics of the transient solutions have been successfully addressed. The optimum of the complex‐valued artificial dispersion has been derived for one‐dimensional problems. In multidimensional cases the streamline upwind scheme is modified by adding complex‐valued artificial dispersion along the streamline. Within this numerical scheme the grid orientation problems have been successfully treated. The limitations on the cell Peclet number and on the Courant number were greatly relaxed. Both one‐dimensional and two‐dimensional numerical examples are used to illustrate applications of this technique.

Laplace-transform approach to the quantal Coulomb problem

A complete solution of the quantal non relativistic Coulomb problem is derived using the most elementary properties of Laplace transform.

Electrocrystallisation: Multilayer formation and potentiostatic transients

Viewing the multilayer formation as a cascade process, the problem of potentiostatic transients is solved. The Laplace Transform approach used here, permits an easy, exact evaluation of steady state current i(t→∞), with the help of Tauber's theorem. The conjecture, based on simulation studies, by Armstrong and Harrison, is shown to be in satisfactory agreement with our exact, analytical results. A theoretical procedure for recovering the transient over the entire time domain is also indicated.

Two-Dimensional Wave Propagation in Realistic Viscoelastic Materials

A solution method using realistic (broad-band) viscoelastic response characteristics covering approximately ten decades of logarithmic time is presented for wave propagation in two-dimensional geometries. A Dirichlet series representation of the viscoelastic mechanical properties and a numerical collocation inversion procedure overcome many of the computational difficulties associated with the Laplace-transform approach to dynamic linear viscoelasticity and also afford a complete time solution. Stress distributions are given for two complementary cases, (a) the viscoelastic half-space subjected on the upper surface to the steady motion of a step pressure load moving supersonically relative to any wave speed in the material, and (b) the semi-infinite thin plate similarly loaded on its edge.