We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in R 3 . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation DW = (Df /f )W in R 3 , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell's equations with only variable permeability.

For the one‐dimensional Schrödinger equation with short‐range potential on a half‐line x>0, the knowledge of the Jost solution e(ρ,x)∼eiρx, Imρ ≥ 0, x→∞ allows one to solve corresponding spectral problems. In the present work, a new series representation for e(ρ,x) is derived with the aid of the Levin formula for the Jost solution and a recently proposed Fourier‐Laguerre series expansion of the integral kernel from the Levin formula. The representation for e(ρ,x) has the form efalse(ρ,xfalse)=eiρxfalse(1+∑n=0∞znbnfalse(xfalse)false), where, for the coefficients bn(x), a simple recurrent integration procedure is obtained and the parameter z:=()12+iρfalse/()12−iρ belongs to the unit disk. An analogous representation is derived for the derivative of the Jost solution as well. With the aid of the series representations, numerical solution of the classical spectral problem on the half‐line becomes an easy task. Indeed, computation of the eigenvalues reduces to finding zeros of a polynomial for z∈(−1,1). For computing corresponding normalizing constants, a simple formula is derived. Moreover, computation of the (usually) most difficult spectral characteristics, the spectral density function (or its derivative) for all positive values of the spectral parameter reduces to calculation of an easily computable function along the unitary upper half‐circle. Numerical illustrations of the efficiency of the computational method based on the obtained results are presented.

The three-dimensional Hilbert transform takes scalar data on the boundary of a domain Ω ⊆ R 3 and produces the boundary value of the vector part of a quaternionic monogenic (hyperholomorphic) function of three real variables, for which the scalar part coincides with the original data. This is analogous to the question of the boundary correspondence of harmonic conjugates. Generalizing a representation of the Hilbert transform H in R 3 given by T. Qian and Y. Yang (valid in R n ), we define the Hilbert transform H f associated to the main Vekua equation DW = (Df /f )W in bounded Lipschitz domains in R 3 . This leads to an investigation of the three-dimensional analogue of the Dirichlet-to-Neumann map for the conductivity equation.

The inverse Sturm-Liouville problem on a half-line is considered. With the aid of a Fourier-Legendre series representation of the transmutation integral kernel and the Gel'fand-Levitan equation, the numerical solution of the problem is reduced to a system of linear algebraic equations. The potential q is recovered from the first coefficient of the Fourier-Legendre series. The resulting numerical method is direct and simple. The results of the numerical experiments are presented.

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.