Abstract In this paper, the equation formed by a combination of the equations AKNS ⁡ ( + 1 ) \operatorname{AKNS}(+1) and AKNS ⁡ ( - 1 ) {\operatorname{AKNS}(-1)} was integrated using the inver problem method for the self-adjoint periodic Dirac operator in the class of periodic infinite-gap functions. In addition, an infinite system of Dubrovin differential equations that represents evolution of spectral data of the Dirac operator is derived and it is proved that the Cauchy problem for the system of Dubrovin differential equations has a unique solution in the class of three times continuously differentiable periodic infinite-gap functions.

Abstract This manuscript investigates the identification problem for a second-order abstract neutral functional differential equation incorporating both instantaneous and non-instantaneous impulses in a Banach space. We establish distinct results for two fundamental cases based on the problem’s singularity. For the non-singular case, we prove the existence and uniqueness of a mild solution using perturbation theory, the properties of strongly continuous cosine families, and duality in functional analysis. For the singular case, we propose an iterative variable-replacement framework that successively modifies the original problem until a non-singular form is obtained, enabling the derivation of a unique mild solution via a cyclic vector. Our theoretical findings are validated with a concrete example.

Abstract The paper presents an approach to the scattering problem on the half-line for a class of dynamical systems involving two types of waves that propagate at different velocities and interact within a non-stationary medium. The scattering operator on the half-line is defined, and its Volterra property is established. A procedure is then proposed for solving the inverse scattering problem, namely, recovering the time-dependent potential from the scattering operator. This work is inspired by M. I. Belishev et al. (1997), who addressed the reconstruction of a spatially varying potential of the dynamical system coupling two one-dimensional wave equations from scattering data.

Abstract In this work, a unified approach for evaluating the European Put and Call options as well as the Barrier options from exponential moments, the moment recovered-Laplace transform (MR-LT) inversion method is introduced. (see also [24] and [25]). In addition, the insurance stop-loss premium and the bivariate probability density function and corresponding tail distribution of aggregate claims are approximated. Several examples are considered to illustrate the accuracy of newly defined approximations.

Abstract In this paper, an ill-posed problem of multi-term time-fraction diffusion equation with unknown initial values is considered. This is usually done by g := u ⁢ ( ⋅ , T ) {g:=u(\,\cdot\,,T)} for some T > 0 {T>0} to find u ⁢ ( x , t ) {u(x,t)} for 0 ≤ t < T {0\leq t<T} which is so-called backward problem. For convenience, we will label this problem as ( P t ) {(P_{t})} . Through analysis, we know that ( P t ) {(P_{t})} is well posed for 0 < t < T {0<t<T} , but ill-posed for t = 0 {t=0} that is different from classical backward heat conduction problem. Therefore, a novel identification scheme of stable approximate solutions of ill-posed inverse problem ( P 0 ) {(P_{0})} is obtained by regularization family { R t : 0 < t < T } {\{R_{t}:0<t<T\}} , and the error estimates are analyzed under an a-priori and an a-posteriori regularization parameter choice rule and source conditions. Finally, some numerical examples are given to verify the effectiveness of the proposed method in one- and two-dimensional cases.

Abstract This article provides an overview of Jan Boman’s illustrious seventy year career as an approximation theorist, microlocal analyst, and integral geometer. We will include his main mathematical themes and some personal observations.

Abstract This paper is devoted to identifying source term and initial value simultaneously in a time-fractional Black–Scholes equation, which is an ill-posed problem. The inverse problem is transformed into a system of unbounded operator equation system, and conditional stability is established under certain source conditions. We propose a stable approximation method to solve the problem, error estimates by rules of a priori and a posteriori regularization parameter selection are derived respectively. Numerical experiments are presented to validate the effectiveness of the proposed regularization method.

Abstract The control problem for a nonlinear reaction–diffusion–convection equation is studied. The role of multiplicative controls is played by the diffusion coefficient, by the mass transfer coefficient in the Robin boundary condition and also by the velocity vector. Boundary and distributed controls are also used. The solvability of the extremum problem is proved under minimal conditions on multiplicative controls. For a specified reaction coefficient optimality systems are derived for control problems. On the basis of the analysis of these systems the bang–bang principle for a distributed control is established; additionally, the local stability estimates of optimal solutions are derived with respect to small perturbations of both cost functionals and of one of specified functions from the boundary value problem.

Abstract This paper establishes a stability estimate for the inverse problem of identifying the time-dependent source function in parabolic equations, where coefficients vary with space and time, by employing the Carleman estimate technique.