Integration Of Differential Equations Research Articles

A Two-Dimensional Map Derived From An Ordinary Difference Equation of mKdV and Its Properties

The discrete modified Korteweg–de Vries (mKdV) is a class of discrete integrable systems that may be distinguished as integrable partial difference equations (PΔE) and integrable ordinary difference equations (OΔE). By considering traveling wave solutions, the OΔE mKdV can be obtained from PΔE mKdV. Meanwhile, a mapping can be constructed from an OΔE mKdV. In this paper, we will focus on producing a new map using a process (replacement), the interchange of a single parameter, and an integral and investigate its properties.

On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

A New Modification of Newton Method with Cubic Convergence

Newton’s method is used to find the roots of a system of equations f (x) = 0. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton’s method was proposed by McDougall and Wotherspoon [1] with an order of convergence of 1+ √2. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier [2]. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.

Existence of a solution of fractional differential equations using the fixed point technique in extended $ b $-metric spaces

<abstract><p>The purpose of the present paper is to prove some fixed point results for cyclic-type operators in extended $ b $-metric spaces. The considered operators are generalized $ \varphi $-contractions and $ \alpha $-$ \varphi $ contractions. The last section is devoted to applications to integral type equations and nonlinear fractional differential equations using the Atangana-Bǎleanu fractional operator.</p></abstract>

On the Use of Perturbation Theory in Eigenvalue Problems

The importance of perturbation theory in many fields is very clear through almost a century or even more. Its importance was exemplified in solving many problems in physics and other applied fields. A great deal of applications arose in dealing with eigenvalue problems especially in quantum mechanics in conjunction with the field of atomic physics. Accordingly, it came to our mind to write a brief review article on the subject. At the beginning, we give some important definitions to do with various eigenvalue problems; then we introduce concepts that have to do with perturbation theory and the techniques used in such a theory, beginning with the algebraic perturbation theory giving a good number of examples from the literature on the use of the theory in solving integral equation, algebraic equations and differential equations. Few applications are then given in applied fields such as classical mechanics, quantum mechanics and fluid mechanics. Finally, a concluding discussion is given which is related to the use of the theory.

Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.

State-dependent graviton noise in the equation of geodesic deviation

We consider an equation of the geodesic deviation appearing in the problem of gravitational wave detection in an environment of gravitons. We investigate a state-dependent graviton noise (as discussed in a recent paper by Parikh,Wilczek and Zahariade) from the point of view of the Feynman integral and stochastic differential equations. The evolution of the density matrix and the transition probability in an environment of gravitons is obtained. We express the time evolution by a solution of a stochastic geodesic deviation equation with a noise dependent on the quantum state of the gravitational field.

High-level model based calibration technique design for SAR ADC

This article presents a method of developing and debugging algorithms for mixed-signal schemes using mathematical model. SAR ADC is used as an example to present the method. A divided capacitor array with is used in the mathematical model. The Laplase transform is used to replace differential integral equations with algebraic ones. The input dependence of the comparator on the switches in the capacitor array is the basis of the mathematical model. There are two additional models – capacitance mismatch model, which was performed by randomly adding the value of the uniform distribution from -0.01 to 0.01 unit of the smallest capacitance, and noise model, which was performed as randomly changing comparator offset. As a result, there is a mathematical model of SAR ADC with event-driven architecture. Switching of K0 sample switch and bit switches leads to changing of circuit state. Mismatch in the capacitor array during chip manufacturing leads to deterioration of key characteristics of ADC – signal-to-noise ratio and spurious-free dynamic range. There is an algorithm, that can calculate the capacitors mismatch and correct the output code. Such an algorithm shows the efficiency under the conditions of capacitor mismatch and comparator input noise. With this algorithm, the signal-to-noise ratio has been increased by 7.3 dB, which is equivalent to 1.2 bit ENOB. The spurious-free dynamic range has been increased by 16.8 dB. Using the mathematical model when designing and debugging calibration algorithms speeds up these processes.

UNAMBIGUOUS SOLVABILITY OF A PARTICULAR CASE OF SYSTEMS OF INTEGRODIFFERENTIAL EQUATIONS WITH A PULSED KAEV DISTANCE CONTAINING A PARAMETER

We consider a special case of systems of integro-differential equations with a momentum boundary condition containing a parameter when the derivative of the desired function is contained in the right side of the equation. By integrating in parts, an integro-differential equation with a pulsed boundary condition is reduced to a loaded integrodifferential equation with a pulsed boundary condition. it is given in the system of integral-differential equations with impulse boundary conditions parametrically loaded. Then, by entering new parameters, as well as passing to new variables based on these parameters, the problem is reduced to an equivalent problem. Switching to new variables makes it possible to get the initial conditions for the equation. Based on this, the solution of the problem is reduced to solving a special Cauchy problem and a system of linear equations. Using the fundamental matrix of the main part of the differential equation, an integral equation of the Volterra type is obtained. The method of sequential approximation determines the unique solution of the integral equation. Based on this, we find a solution to the special Cauchy problem and put it in the boundary conditions. On the basis of the obtained system of linear equations, necessary and sufficient conditions for an unambiguous solution of the initial problem are established.

On some theory of monostable and bistable pure birth-jump integro-differential equations

We study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics, fire propagation, and cancer cell dynamics. We contrast the dynamics of this equation with those of the classical reaction-diffusion equation, where the reaction term models either logistic growth or a strong Allee effect. Recent evidence of an Allee effect has been found in plant dynamics during the germination process (due to seed predation) but not in the generation of seeds. This motivates where the Allee effect is included in our model. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species, which are analogous to those of the classical reaction-diffusion equation. A key finding is that in some cases a population which is initially below the Allee threshold in some area, even if small, will actually survive. This is in contrast to solutions of the classical reaction-diffusion with the same initial data. Another difference of note is the lack of regularization and an infinite number of discontinuous equilibrium solutions to the birth-jump model.

On Sobolev rough paths

We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity α and integrability p provided α>1/p.

Analytical Methods for Nonlinear Evolution Equations in Mathematical Physics

In this article, we will apply some of the algebraic methods to find great moving solutions to some nonlinear physical and engineering questions, such as a nonlinear (1 + 1) Ito integral differential equation and (1 + 1) nonlinear Schrödinger equation. To analyze practical solutions to these problems, we essentially use the generalized expansion approach. After various W and G options, we get several clear means of estimating the plentiful nonlinear physics solutions. We present a process like-direct expansion process-method of expansion. In the particular case of W′=λG, G′=μW in which λ and μ are arbitrary constants, we use the expansion process to build some new exact solutions for nonlinear equations of growth if it fulfills the decoupled differential equations.

Identification of non-local continua for lattice-like materials

The paper is focused on the dynamic homogenization of lattice-like materials with lumped mass at the nodes to obtain energetically consistent models providing accurate descriptions of the acoustic behavior of the discrete system. The equation of motion of the Lagrangian one-dimensional lattice is transformed according to a unitary approach aimed to identify equivalent non-local continuum models of integral-differential and gradient type, the latter obtained through standard or enhanced continualization. The bilateral Z-transform of the difference equation of motion of the lattice, mapped on the unit circle, is matched to the governing integral-differential equation of the equivalent continuum in the transformed Fourier space, which has the same frequency band structure as the Lagrangian one. Firstly, it is shown that the approximation of the kernels via Taylor polynomials leads to the differential field equations of higher order continua endowed with non-local constitutive terms. The field equations derived from such approach corresponds to the ones obtained through the so called standard continualization. However, the differential problem turns out to be ill-posed because the non-positive definiteness of the potential energy density of the higher order continuum. Energetically consistent equivalent continua have been identified through a proper mapping correlating the transformed macro-displacements in the Fourier space with a new auxiliary regularizing continuum macro-displacement field in the same space. Specifically, the mapping here introduced has zeros at the edge of the first Brillouin zone. The integral-differential governing equation and the corresponding differential one has been reformulated through an enhanced continualization, that is characterized by energetically consistent differential equations with inertial and constitutive non-localities. Here, the constitutive and inertial kernels of the integral-differential equation exhibit polar singularities at the edge of the first Brillouin zone. The proposed approach is generalized in a consistent way to two-dimensional lattices by using multidimensional Z- and Fourier transforms, a procedure that may be easily extended to three-dimensional lattices. Finally, two examples of lattice-like systems consisting of periodic pre-stressed cable-nets of point mass at the nodes are analyzed. The resulting gradient continuum models provide dispersion curves who are in excellent agreement with those of the Lagrangian systems.

Parameter estimation for grey system models: A nonlinear least squares perspective

Grey system models can be viewed as a special class of dynamic data analysis tools, in which the continuous-time dynamics (differential equations or integral equations) are used to define the implicit regression formula. This communication investigates the parameter estimation of grey system models from noisy observations. Based on the state-space representation, the parameter estimation of dynamics is converted to a nonlinear regression and four optimization criteria are induced under a measurement error environment. Then, nonlinear least squares is utilized to estimate parameters and quantify the uncertainty on the estimates. Under different sample size and noise level combinations, simulations are conducted on the models (with and without exogenous variables) to compare the parameter estimation approaches. The results show that nonlinear least squares has multiple advantages over the conventional integral matching in terms of accuracy and robustness to noise, especially when the observations are irregularly-spaced.

An optimal system of group-invariant solutions and conserved quantities of a nonlinear fifth-order integrable equation

Abstract In this work, we perform Lie group analysis on a fifth-order integrable nonlinear partial differential equation, which was recently introduced in the literature and contains two dispersive terms. We determine a one-parameter group of transformations, an optimal system of group invariant solutions, and derive the corresponding analytic solutions. Topological kink, periodic and power series solutions are obtained. The existence of a variational principle for the underlying equation is proven using Helmholtz conditions and, thereafter, both local and nonlocal conserved quantities are obtained by utilising Noether’s theorem and a homotopy integral approach.

Lerch matrix collocation method for 2D and 3D Volterra type integral and second order partial integro differential equations together with an alternative error analysis and convergence criterion based on residual functions

In this study, second order linear Volterra partial integro-differential equation with two- and three-dimensional are solved by collocation method based on Lerch polynomials. This method is composed of the operational matrix and collocation methods, which are based upon the matrix forms of the Lerch polynomials with the parameter $\lambda$ and Taylor polynomials, and their derivatives and integrals. The approximate solutions of the mentioned equations are investigated in terms of the Lerch polynomials the different values of $\lambda$. Also, to verify the accuracy and efficiency of the present method, an alternative convergence criterion along with error analysis depending on residual function is enhanced. Moreover, the obtained numerical results are compared with other methods and scrutinized by using tables and figures.

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

AbstractIn this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.

Recursion and Hamiltonian operators for integrable nonabelian difference equations

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice.

Relation of Integro-Partial Differential Equations With Delay Effect Based on the Maxwell Equations to the Heaviside and Pocklington Equations

We derive the integro-partial differential equations with delay effect based on the Maxwell equations for a two-line system under the thin-wire approximation and the approximation of neglecting the interwire distance in the delay term. We introduce the normal- and common-mode and write the integro-partial differential equations in terms of these two modes. For the case of the same radius, we show that the equation of the normal-mode reduces to the Heaviside telegraph equation in the partial differential form and the equation of the common-mode becomes the Pocklington equation in the integro-partial differential form. We perform numerical calculations of the coupled integral and partial differential equations in the time domain and compare the results with those of various approximations. We show that the Heaviside telegraph equation is valid for the normal-mode, while the full treatment of the delay term in the integro-partial differential equations is necessary for the common-mode.

МАТЕМАТИЧНА МОДЕЛЬ ХВИЛЬОВИХ ПРОЦЕСІВ У ДВООБМОТКОВИХ ТРАНСФОРМАТОРАХ

A mathematical model has been developed to study the wave processes of two-winding transformers, taking into account the electromagnetic connections between the turns of the winding and between the windings. To solve differential-integral equations in partial derivatives, a method of separation of variables is proposed. References 12, figures 2.