A representation of solutions the one-dimensional Dirac equation is obtained. The are represented as Neumann series Bessel functions. representations shown to be uniformly convergent with respect spectral parameter. Explicit formulas for coefficients obtained via a system recursive integrals. result based on Fourier-Legendre expansion transmutation kernel. An efficient numerical method solving initial-value and problems this approach presented example. can compute large sets eigendata...

A direct method for solving the inverse problem of determining shape cross section a rod is proposed. The based on Neumann series Bessel functions representations solutions Sturm-Liouville equations. first coefficient representation sufficient recovery unknown function. system linear algebraic equations finding this obtained. proposed leads to an efficient numerical algorithm.

A new representation for a regular solution of the perturbed Bessel equation form is obtained. The represented as Neumann series functions uniformly convergent with respect to . For coefficients series, explicit direct formulas are obtained in terms systems recursive integrals arising spectral parameter power (SPPS) method, well convenient numerical computation recurrent integration formulas. result based on application several ideas from classical transmutation (transformation) operator...

We solve the following problem. Let q1 be a continuous complex-valued potential of stationary Schrödinger operator defined on segment [ − a, a] and q2 Darboux transformed operator, that is , where f nonvanishing solution equation . Suppose transmutation T1 known such for any u ∈ C2[ a]. Find an analogous It well operators can realized in form Volterra integral with continuously differentiable kernels. Given kernel K1 T1, we find K2 T2 closed terms K1. As corollary, interesting commutation...

We give an overview of recent developments in Sturm-Liouville theory concerning operators transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding a variety problems related analytic form is attractive feature SPPS method. It based on computation certain systems recursive integrals. Considered as families functions these are complete L22-space result be images nonnegative integer powers...

This article aims to study the Coulomb gas model over d-dimensional p-adic space. We establish existence of equilibrium measures and Γ-limit for energy functional when number configurations tends infinity. For a cloud charged particles confined into unit ball, we compute measure minimum its functional. The is continuum limit minus hierarchical Hamiltonian attached spin glass with coupling.

We consider two main inverse Sturm-Liouville problems: the problem of recovery potential and boundary conditions from spectra or a spectral density function. A simple method for practical solution such problems is developed, based on transmutation operator approach, new Neumann series Bessel functions representations solutions Gelfand-Levitan equation. The allows one to reduce directly system linear algebraic equations, that recovered first element vector. prove stability show its numerical...

A method for practical realization of the inverse scattering transform Korteweg–de Vries equation is proposed. It based on analytical representations Jost solutions and integral kernels transformation operators obtained recently. The have form functional series in which first coefficient plays a crucial role both solving direct problems. problem reduces to computation number coefficients following simple recurrent integration procedure with posterior calculation data by well known formulas....

A complete family of solutions for the one-dimensional reaction-diffusion equation, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>-</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo...

Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on SPPS analysis of graded-index optical fibers. The characteristic equation eigenvalue problem calculation guided modes obtained in analytical form terms SPPS. Truncation consideration this way approximate gives us simple efficient numerical problem. Comparison with results by other...

A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations the form $$(pu')'+qu=u\sum_{k=1}^{N}\lambda^{k}r_{k}$$ is obtained. It allows one to write a general solution equation as in terms $\lambda$. The coefficients are given recursive integrals involving particular $(pu_{0}')'+qu_{0}=0$. convenient provides an efficient numerical method solving corresponding initial value, boundary value and problems. special case considered arises relation with...

A numerical method for free boundary problems the equation urn:x-wiley:mma:media:mma5483:mma5483-math-0001 is proposed. The based on recent results from transmutation operators theory allowing one to construct efficiently a complete system of solutions above equation, generalizing heat polynomials. corresponding implementation algorithm presented.