Journal of Mathematical Chemistry

An inverse problem framework for constructing reaction systems with prescribed properties is presented. Kinetic transformations are defined and analysed as a part of the framework, allowing an arbitrary polynomial ordinary differential equation to be mapped to the one that can be represented as a reaction network. The framework is used for construction of specific two- and three-dimensional bistable reaction systems undergoing a supercritical homoclinic bifurcation, and the topology of their phase spaces is discussed.

This paper presents a high-order computational scheme for numerical solution of a time-fractional diffusion equation (TFDE). This scheme is discretized in time by means of L1-scheme and discretized in space using a compact finite difference method. Stability analysis of the method is discussed. Further, convergence analysis of the present numerical scheme is established and we show that this scheme is of $$O({\Delta t}^{2-\alpha }+{\Delta x}^{4})$$ convergence, where $$\alpha \in (0,1)$$ is the order of fractional derivative (FD) appearing in the governing equation and $$\Delta t$$ and $$\Delta x$$ are the step sizes in temporal and spatial direction, respectively. Three numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by finite element method and B-spline method. Comparison reveals that the proposed method is fast convergent and highly accurate. Moreover, the effect of $$\alpha$$ on the numerical solution of TFDE is investigated. The CPU time of the present method is provided.

In this paper some new eighth algebraic order symmetric eight-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Based on this formula, the calculation of free parameters is done in order the phase-lag to be minimal. The new methods have better stability properties than the classical one. Numerical illustrations on the radial Schrödinger equation indicate that the new method is more efficient than older ones.

The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically, thus a numerical scheme has been applied.

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow in combinatorial complexity as a function of edge or vertex coloring partitions.

A new model of the Belousov-Zhabotinskii reaction is developed. It describes bistable behavior of the reaction. For this reaction -diffusion system existence results are proved. The critical radius of a nucleus is defined and studied by numerical methods.

A detailed spectral map for the fast-time instability in Liñán’s diffusion-flame regime is presented in order to clarify the origin of two bifurcations of co-dimension 2, causing the transitions from cellular to uniform-oscillatory instability and from uniform-oscillatory to traveling instability. The role of the real and continuous essential spectrum is found to be pivotal in understanding both transitions. Particular attention is paid to the spectral characteristics in the stable parametric regions, where the interaction with the essential spectrum leads to these transitions. When the Lewis number is increased above unity from below, the discrete real spectrum disappears by submerging below the essential spectrum, and the discrete complex spectrum emerges instead, eventually leading to uniform-oscillatory instability. The transition from uniform-oscillatory to traveling instability, associated with the Bogdanov–Takens bifurcation, involves a phenomenon called gap spectrum. For Lewis numbers slightly greater than unity and Damköhler numbers sufficiently large, the discrete complex spectrum intersects the plane corresponding to the essential spectrum, resulting in a gap in the discrete spectrum for small wave numbers. The discrete complex gap spectrum exhibits a local maximum as the parameter values are modified to approach the Hopf bifurcation boundary. The gap in the discrete complex spectrum disappears and traveling instability emerges when crossing the Hopf bifurcation boundary.

Anm 5-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.