Types Of Differential Equations Research Articles

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.

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.

N-Best kernel approximation in reproducing kernel Hilbert spaces

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of n-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of n-best rational approximation for the Hardy space and a recent result of n-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

Stability of Some Anticipating Semilinear Stochastic Differential Equations of Skorohod Type

AbstractIn the present paper, we study different types of stability of the solution of a semi-linear anticipating stochastic differential equation driven by a Brownian motion, with a random variable as initial condition. The involved stochastic integral is the Skorohod one. Being the initial condition random, we need to redefine the stability concepts. The new stability criteria depend on the derivative of the initial condition in the Malliavin calculus sense.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Identification of sources in a boundary value problem for Benney-Luke type differential equation with integral conditions

Identification of sources in a boundary value problem for Benney-Luke type differential equation with integral conditions

On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

Oscillatory Behavior of Higher Order Nonlinear Mixed Type Difference Equations With a Nonlinear Neutral Term

This paper discusses higher order nonlinear neutral mixed type difference equations of the form
 Δ^{m}[x(n)+p(n)h(x(σ(n)))]+q(n)f(x(τ(n)))=0, n=0,1,2,…,
 where (p(n)), (q(n)) are sequences of nonnegative real numbers, h, f:R→R are continuous and nondecreasing with uh(u)>0, uf(u)>0 for all u≠0, and (σ(n)) and (τ(n)) are sequences of integers such that
 lim_{n→∞}τ(n)=lim_{n→∞}σ(n)=∞.
 In general, we will examine the oscillatory behavior of the solutions for the above equation. Especially, when m is even, the result obtained here complements studies related to the oscillation of the above equation. In addition, examples showing the accuracy of the results are given.

On meromorphic solutions of Malmquist type difference equations

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form \(f(z+1)^n=R(z,f)\) \((\dagger)\) that have transcendental meromorphic solutions, where \(R(z,f)\) is rational in both arguments. In this paper, we first complete the classification for the case \(\deg_{f}(R(z,f))=n\) of \((\dagger)\) by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on \((\dagger)\). Then, we study the relations between \((\dagger)\) and its differential counterpart \((f')^n=R(z,f)\). We show that most autonomous equations, singled out from \((\dagger)\) with \(n=2\), have a natural continuum limit to either the differential Riccati equation \(f'=a+f^2\) or the differential equation \((f')^2=a(f^2-\tau_1^2)(f^2-\tau_2^2)\), where \(a\neq 0\) and \(\tau_i\) are constants such that \(\tau_1^2\neq\tau_2^2\). The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.

Large Fractional Linear Type Differential Equations

This paper aims to handle some types of fractional differential equations with a fractional-order values β>1. In particular, we propose a novel analytical solution called an atomic solution for certain fractional linear type differential equations as well as for some other types of partial differential equations with fractional-order values exceeding one. Some examples are provided to validate our findings.