Related papers: Complex Dynamics of a Second Order Rational Differ…
The second order partial difference equation of two variables $ \CD u:= A_{1,1}(x) \Delta_1 \nabla_1 u + A_{1,2}(x) \Delta_1 \nabla_2 u + A_{2,1}(x) \Delta_2 \nabla_1 u + A_{2,2}(x) \Delta_2 \nabla_2 u & \qquad \qquad \qquad \qquad + B_1(x)…
For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated…
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…
We give a complete description of the qualitative behavior of the second-order rational difference equation #166. We also establish the boundedness character for the rational system in the plane #(8,30).
We study various dynamical aspects of systems possessing a first order phase transition in their phase diagram. We isolate three qualitatively distinct types of theories depending on the structure of instabilities and the nature of the low…
The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and…
Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma…
The transformation of the Nth- order linear difference equation into a system of the first order difference equations is presented. The proposed transformation gives possibility to get new forms of the N-dimensional system of the first…
Complex-linearization of a class of systems of second order ordinary differential equations (ODEs) has already been studied with complex symmetry analysis. Linearization of this class has been achieved earlier by complex method, however,…
Lie group analysis of the difference equations of the form \begin{align*} x_{n+1} =\frac{x_{n-4}x_{n-3}}{x_{n}(a_n +b_nx_{n-4}x_{n-3}x_{n-2}x_{n-1})}, \end{align*} where $a_n$ and $b_n$ are real sequences, is performed and non-trivial…
We study the following system of two rational difference equations x_n=({\beta}_k x_(n-k)+{\gamma}_k y_(n-k))/(A+\Sigma_(j=1)^l[B_j x_(n-j) ]+\Sigma_(j=1)^l[C_j y_(n-j) ]), n \in N, y_n=({\delta}_k x_(n-k)+\in_k…
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
We introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex H\"older space $C^{n+1,\alpha}$, with $0\leq n\in\mathbb{Z}$ and…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
We exhibit an alternative method for solving inhomogeneous second--order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis…
Applying Zeilberger's algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan's constant with rational coefficients, we obtain a second-order difference equation for…
In this research paper, we provide a concise overview of fractal calculus applied to fractal sets. We introduce and solve a second $\alpha$-order fractal differential equation with constant coefficients across different scenarios. We…
For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the…