Types Of Differential Equations Research Articles

THE EXISTENCE AND UNIQUENESS THEOREMS FOR STOCHASTIC DIFFERENTIAL-DIFFERENCE HYBRID SYSTEMS

A stochastic differential-difference hybrid system is a system of interacting variables whose dynamics are described by stochastic differential equations for some of them and difference equations for others. Systems with two types of difference equations are examined: first, a difference equation in the form of a process involving the multiplicative Wiener process, and second, a difference equation with delay. The existence and uniqueness theorems for both systems have been proven. The basic conditions on the system’s parameters are local Lipschitz conditions and linear growth order.

Measuring and modeling the solubility of sulfasalazine in supercritical carbon dioxide to select methods for producing nanoparticles

In the present study, the solubility of sulfasalazine in carbon dioxide was investigated at temperatures ranging from 313 K to 343 K and pressures ranging from 12 to 30 MPa. The experimentally determined molar solubilities of sulfasalazine in ScCO2 were found to be in the range of 4.08 × 10− 5 to 8.61 × 10− 5 at 313 K, 3.54 × 10− 5 to 11.41 × 10− 5 at 323 K, 3.04 × 10− 5 to 13.64 × 10− 5 at 333 K, and 2.66 × 10− 5 to 16.35 × 10− 5 at 343 K. The solubility values were correlated via a number of different types of equations, such as semi-empirical correlations, the Peng-Robinson, the PC-SAFT equation, and the regular solution. Furthermore, the findings demonstrate that semi-empirical, equation of state models, and the regular solution model possess the capability of precisely determining the solubility. Moreover, the solubility magnitude suggests that the gas anti-solvent method may be a viable approach for nanoparticle production.

An asymptotic result for linear nonhomogeneous mixed type differential equation

We consider a nonhomogeneous linear mixed type differential equation with variable coefficients and establish an asymptotic result for the solutions. Our result is obtained by the use of a solution of the so-called generalized characteristic equation of the corresponding homogeneous linear mixed type differential equation.

Optimal control governed impulsive neutral differential equations

We derive Pontryagin’s Maximum Principle for optimal control problems characterized by nonlinear impulsive neutral type differential equations. Our method utilizes the Dubovitskii-Milyutin theory, assuming that the linear variational impulsive differential equation along the optimal solution is exactly controllable. This principle offers necessary conditions for identifying optimal solutions.

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Meromorphic solutions of Bi-Fermat type partial differential and difference equations

Finite orbits of the braid group actions

We study the finite orbits of the braid group Bn action on the space of n×n upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on 3×3 matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic n×n matrices to be useful to finding solutions of higher order Painleve type differential equations.

A note on the telegraph type differential equation with time involution

In the present paper, the initial value problem for the telegraph type involutory in t second order linear partial differential equation with damping term is investigated. The equivalent initial value problem for the fourth order partial differential equations to the initial value problem for this second order linear partial differential equations with involution and damping term is obtained. Applying the operator tools, the stability estimates for the solution and its first and second order derivatives of this problem are established.

New algorithm based on Hermite wavelets and Green’s function to solve Thomas-Fermi type differential equations

This article introduces a novel method using Hermite wavelets and Green’s function to solve Thomas-Fermi-type equations with specific boundary conditions. The equation is first transformed into an integral form and then expanded using Hermite wavelets to create an algebraic expression. The collocation technique generates a set of nonlinear equations, solved iteratively by methods like Newton-Raphson. Several examples demonstrate the method’s effectiveness and accuracy. It is useful for a range of linear and nonlinear differential equations in various fields. Comparative analysis shows it is more accurate and computationally efficient than traditional methods. All computations are performed using MAT LAB.

A nonlinear instability theory for a wave system inducing transition in spiral Poiseuille flow

This paper is on the transition scenario of the class of spiral Poiseuille flows that results from the onset, propagation and evolution of disturbances according to mechanisms of Tollmien-Schlichting, and Taylor, acting simultaneously. The problem is approached from the fundamental point of view of following the growth of initially infinitesimally small disturbances into their nonlinear stage when the effect of Reynolds stresses makes itself felt. To this end a set of Generalised Nonlinear Orr–Sommerfeld, Squire and Continuity Equations is set up that enables accounting for effects of growth of initially infinitesimally small disturbances into nonlinearities through a rational iteration scheme. The present proposal closely follows the method put forth for this pupose in 1971 by Stuart and Stewartson in their seminal papers on the influence of nonlinear effects during transition in the bench-mark flows of the class of spiral Poiseuille flows; which are the plane-walled channel flow and the flow in the gap between concentric circular cylinders (Taylor instability).The basic feature of the proposed method is the introduction of an Amplitude Parameter and of a slow/long- scale variable through which the effects of growing disturbances are accounted for within the framework of a rational iteration scheme. It is shown that the effect of amplified disturbances is capturable, as in the bench-mark flows, by a Ginzburg–Landau type differential equation for an Amplitude Function in terms of suitably defined slow/long-scale variables. However, the coefficients in this equation are numbers that depend upon the flow parameters of the spiral Poiseuille flow, which are a suitably defined Reynolds Number, the Swirl Number, and the geometric parameter of transverse curvature inherent in the flow geometry. The Ginzburg–Landau equation derived hints at the drastic changes in flow pattern that the spiral Poiseuille flow in transition may undergo, as its Swirl Number is taken from very small to very large values.

On Meromorphic Solutions of the Fermat Type Difference Equations

On Meromorphic Solutions of the Fermat Type Difference Equations

Existence and Uniqueness of a Second Order Difference Equation of Accretive Type in 2-Banach Spaces

In this paper, we investigate the existence and uniqueness of solutions to a homogeneous second order difference inclusion of accretive type in 2-Banach spaces using expansive mappings and 2-Banach contraction mapping.

Bethe/Gauge correspondence for AN spin chains with integrable boundaries

We continue the survey initiated in [1] to explore the Bethe/Gauge correspondence between supersymmetric SO/Sp gauge theories in 2d/3d/4d and open spin chain with integrable boundaries. We collect the known Bethe ansatz equations of different types of spin chains with general boundaries that have been analyzed in the literature, and compare them with the vacua equations of the quiver gauge theories. It seems that not all the vacua equations of quiver gauge theory with BCD-type gauge groups can be realized as some known Bethe ansatz equations of integrable spin chain models.

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

On entire solutions of Fermat type difference and kth order partial differential difference equations in several complex variables

ON THE EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NON-HOMOGENEOUS SECOND-ORDER DIFFERENCE EQUATION OF ACCRETIVE TYPE IN 2-BANACH SPACES

In this study we present the existence and uniqueness of a non-homogeneous second order difference equation of accretive type in linear 2-Banach spaces. We use the 2-norm contraction mapping and its fixed point to establish our results.

Mathematical analysis of a randomized method for fractional Carathéodory type equation with time‐irregular coefficients

AbstractIn this work, we present a novel error analysis for fractional Carathéodory type differential equations with time irregular coefficients. It is based on the filtered probability space, with Banach spaces and then discretized using the randomized numerical method, which is a prototype derived from the Monte Carlo method. We derive error estimates where these estimates also under the low regularity that the function is not assumed to be differentiable. The convergence order is also studied. Finally, we present numerical examples that validate the theoretical conclusions, as well as to highlight the usefulness of our technique and the accuracy of error analysis.

A design proposal of finite‐time H∞ controller for stochastic mean‐field systems

AbstractA design proposal of finite‐time controller for stochastic mean‐field systems (MFSs), which are described by a Itô type differential equation, is presented in this paper. Firstly, based on control theory and finite‐time control theory, three sufficient conditions, which ensure the MFS satisfies stochastic finite‐time state feedback stabilization, are gained. Secondly, by decoupling correlation matrix inequalities, the state feedback controller (SFC) gains are obtained. Finally, the simulation results demonstrate that the large system of interacting particles can be approximated by MFS under certain conditions, and the controller design schemes proposed are feasible.

A Novel Approach with Comparative Computational Simulation of Analytical Solutions of Fractional Order Volterra Type Integral Equations

The main purpose of this article is to reconsider the theory of variational iteration method and embed it with Laplace transform to construct the conjoined technique variational iteration Laplace transform method. And then, by using the proposed method we solve a class of integral equations, especially fractional order Volterra type integro‐partial‐differential equations occurring in different fields of natural and social sciences. For this, we first turn to some prelude concepts and different types of equations especially integral equations of different kinds from the literature. Then, we will present step by step the embedded technique. Next, some fractional order model integral equations will be tackled using the proposed method along with some existing techniques, like the Adomian decomposition method and successive approximations method. The efficacy and authentication of the proposed technique will be analysed with the help of the obtained results and graphical exhibitions of the particular solved examples. Finally, use of the initiated method in the future for different types of equations occurring in different fields of science will be presented.

Existence and controllability results of semilinear Sobolev type difference equation

Existence and controllability results of semilinear Sobolev type difference equation

Oscillation criterion for Euler type half‐linear difference equations

We consider general classes of Euler type linear and half‐linear difference equations, which are conditionally oscillatory. Applying the adapted Riccati technique, we improve known oscillation criteria for these equations. More precisely, our presented main criterion is the full oscillatory counterpart of a non‐oscillation criterion. Thus, in this paper, we enlarge the set of conditionally oscillatory Euler type difference equations. We highlight that our results are new even for linear equations with periodic coefficients. This fact is documented by simple examples of such equations at the end of this paper.

Zeros transfer for recursively defined polynomials

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.