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Without taking into account external friction, small transverse vibrations of a viscoelastic pipeline of unit length are described for nonnegative values of time in dimensionless variables by an integro-differential equation with hinged conditions at the ends and with initial conditions. The solution of this equation can be written in terms of an operator semigroup. In the present paper, we establish that this equation generates a semigroup that is analytic in some sector of the right half-plane. [ABSTRACT FROM AUTHOR]

Background: Numerical methods are used to solve differential equations, but few are effective for nonlinear ordinary differential equations (ODEs) of order higher than one. This paper proposes a new method for such ODEs, based on Taylor series expansion. The new method is a second-order method for second-order ODEs, and it is equivalent to the central difference method, a well-known method for solving differential equations. The new method is also simple to implement for higher-order differential equations. The proposed technique was applied to solve the Van der Pol and Van der Pol–Duffing equations. It is stable over a wide range of nonlinearity and produces accurate and reliable results. For the self-excitation Van der Pol equation, the proposed technique was applied with different values of nonlinear damping. Results: The results were compared with those obtained using the ODE15s solver in MATLAB. The two sets of results showed excellent agreement. For the forced Van der Pol–Duffing equation, the proposed technique was applied with different values of exciting force amplitude and frequency. It was found that for certain conditions, the solution obtained using the proposed technique differed from that obtained using ODE15s. Conclusions: The solution obtained using the proposed technique showed good agreement with the solutions obtained using ODE45 and Runge–Kutta fourth order. The results show that the proposed approach is very simple to apply and produces acceptable error. It is a powerful and versatile tool for solving of high-order nonlinear differential equations accurately. [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical theory of normal forms for continuous and difference dynamical systems to dynamic equations on discrete time scales. As consequences of the well known results from the theory of analytic differential equations, we obtain some versions of the Poincaré and Siegel theorems for dynamic equations on discrete time scales. Using these results and known results on the stability of dynamic equations on time scales, we obtain some stability results for the nonlinear dynamic equations. We also prove some results on generic properties of bifurcation curves and the saddle-node bifurcation for one-parameter families of dynamic equations on arbitrary time scales. [ABSTRACT FROM AUTHOR]

In this paper, by applying fractional quantum calculus, we study a nonlinear Duffing-type equation with three sequential fractional q-derivatives. We prove the existence and uniqueness results by using standard fixed-point theorems (Banach and Schaefer fixed-point theorems). We also discuss the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of the mentioned Duffing problem. Finally, we present an illustrative example and nice application; a Duffing-type oscillator equation with regard to our outcomes. [ABSTRACT FROM AUTHOR]

In this paper, the He–Elzaki transform method (HEM) is proposed. The method is formulated by combining He's variation iteration method and the modified Laplace transform, known as the Elzaki integral transform. This method is designed to solve the time-fractional telegraph equation that arises in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this method is that the computation of the Lagrange multiplier is not necessarily required through the convolution theorem or integration in recurrence relations. Additionally, to reduce nonlinear computational terms, He's polynomial is determined using the homotopy perturbation method. The proposed method is applied to several examples of nonlinear fractional telegraph equations. The results obtained from these examples demonstrate that the proposed method is an efficient technique that facilitates the process of solving time-fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, with the aid of symbolic computation system Python, and based on the Deep Neural Network, Automatic differentiation and Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimization algorithms, the model parameters of modified Kortewegde Vries (mkdv) equation Kdv–Burger equation and Huxley equation are obtained. We added different amounts of noise to the clean data in experiment and found that with the addition of trace noise, the parameters of the differential equation can also be accurately found. The result indicates that the algorithm has little effect on trace noise and shows better robustness to data noise. The method in this paper has demonstrated the powerful mathematical and physical ability of deep learning to flexibly simulate the physical dynamic state represented by differential equations, which opens the way for us to understand more physical phenomena later and the algorithm may be suitable for the data in practical application. [ABSTRACT FROM AUTHOR]

In this paper, a class of D-operator-based neutral-type Rayleigh equations accompanying mixed delays is introduced, where the damping term is a continuous T-periodic function. Some new sufficient criteria are derived to guarantee the existence and global exponential stability of T-periodic solutions for the addressed equations by utilizing differential inequality techniques. In addition, an illustrative example involving its graphical simulations is afforded to examine the validity of the obtained findings. [ABSTRACT FROM AUTHOR]

The paper studies the problem of optimizing the lowest coefficient understood as a function with values in a Banach space, which enters linearly into an abstract semilinear pseudoparabolic evolutionary differential equation in a Banach space. For this problem, an existence theorem for an optimal control is proved. Due to the nonlinearity of the equation under study, the author uses his previous results on the total preservation of the unique global solvability (on the totally global solvability) and on the estimation of solutions for similar equations. This estimate turns out to be significant in the course of the study. As an example, the Oskolkov's hydrodynamic system of equations is considered. [ABSTRACT FROM AUTHOR]

In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the truncated current algebra, also called generalised Takiff algebra. Our motivation is to produce confluent versions of the celebrated Knizhnik–Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic τ -function. In order to achieve this, we study the confluence cascade of r + 1 simple poles to give rise to a singularity of arbitrary Poincaré rank r as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians. In loving memory of Igor KricheverA great man and outstanding mathematician [ABSTRACT FROM AUTHOR]

Systems of fractional order differential and pseudo-differential equations are used in modeling of various dynamical processes. In the analysis of such models, including stability analysis, asymptotic behaviors, etc., it is useful to have a representation formulas for the solution. In this paper we prove the existence and uniqueness theorems and derive representation formulas for the solution of general systems of fractional multi-order linear pseudo-differential equations through the matrix-valued Mittag-Leffler function. Examples illustrating the obtained results are also provided. [ABSTRACT FROM AUTHOR]

We develop the theory of quantization of spectral curves via the topological recursion. We formulate a quantization scheme of spectral curves which is not necessarily admissible in the sense of Bouchard and Eynard. The main result of this paper and the second part (Iwaki et al. in Voros coefficients for the hypergeometric differential equations and Eynard–Orantin's topological recursion, part II: for the confluent family of hypergeometric equations, preprint; arXiv:1810.02946) establishes a relation between the Voros coefficients for the quantum curves and the free energy for spectral curves associated with the confluent family of Gauss hypergeometric differential equations. We focus on the Weber equation in this article and generalize the result for the other members of the confluent family in the second part. We also find explicit formulas of free energy for those spectral curves in terms of the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

In this paper, the extended direct algebraic (EDA) scheme is utilized for producing more general exact solutions of the Degasperis–Procesi (DP) and Camassa–Holm (CH) equations with Atangana's conformable derivative. By utilizing a suitable travelling wave transformation, change these equations into ordinary differential equations (ODE). New obtained solutions are in the form of singular, dark, bright, and dark-bright soliton. To clarify the physical structures for such forms of equations, some of the attained solutions are expressed as 2D, 3D, and contour graphs. [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical atmospheric Ekman equations to discontinuous differential equations of Ekman flows by considering the eddy viscosity with piecewise form. Local and global analysis for the corresponding discontinuous ODEs are studied respectively. In the section of local analysis, all the possible crossings of solutions are presented over the certain discontinuity linear subspaces associated with a curve. In the section of global analysis, crossings of solutions, bounded and unbounded solutions on dynamics are obtained. [ABSTRACT FROM AUTHOR]

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method. [ABSTRACT FROM AUTHOR]

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green's function of the new mixed fractional differential equations. We calculate the corresponding Green's functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

We consider the first boundary value problem in a rectangular domain for an inhomogeneous fourth-order differential equation with lower-order terms. The uniqueness of a solution of the stated problem is proved. The solution is obtained explicitly using the Green's function constructed. [ABSTRACT FROM AUTHOR]

Many real-world scientific and engineering processes are governed by complex nonlinear interactions, and differential equations are commonly used to explain the dynamics of these complex systems. While the differential equations generally capture the dynamics of the system, they impose a rigid modeling structure that assumes the dynamics of the system are known. Even when some of the dynamical relationships are known, rarely do we know the form of the governing equations. Learning these governing equations can improve our understanding of the mechanisms driving the complex systems. Here, we present a Bayesian data-driven approach to nonlinear dynamic equation discovery. The Bayesian framework can accommodate measurement noise and missing data, which are common in these systems, and accounts for model parameter uncertainty. We illustrate our method using simulated data as well as three real-world applications for which dynamic equations are used to study real-world processes. Supplementary materials accompanying this paper appear online. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the nonlinear fully second-order differential equation with p-Laplacian (ϕ p (x ′)) ′ = f (t , x , x ′) , t ∈ R : = (- ∞ , ∞) , where ϕ p (s) = | s | p - 2 s , p > 1 , function f : R 3 → R is continuous and T-periodic with respect to t. Using the topological transversality method and the barrier strip technique, we obtain new existence results of periodic solutions to the above p-Laplacian equation without growth restrictions. Meanwhile, an application is given for the Rayleigh-type p-Laplacian equation. [ABSTRACT FROM AUTHOR]

Principal and nonprincipal solutions of differential equations play a critical role in studying the qualitative behavior of solutions in numerous related differential equations. The existence of such solutions and their applications are already documented in the literature for differential equations, difference equations, dynamic equations, and impulsive differential equations. In this paper, we make a contribution to this field by examining impulsive dynamic equations and proving the existence of such solutions for second-order impulsive dynamic equations. As an illustration, we prove the famous Leighton and Wong oscillation theorems for impulsive dynamic equations. Furthermore, we provide supporting examples to demonstrate the relevance and effectiveness of the results. [ABSTRACT FROM AUTHOR]

This paper focuses on solving numerically the telegrapher problems in the two-dimensional case, which is a well-known second-order differential equation using a relatively innovative methodology and approach. In terms of approach, we have applied a new robust true mesh-free local approach combining the spectral method and and the Newmark scheme for space and time approximations respectively. Unlike several meshless methods, the proposed mesh-free can be based only on a unidimensional study which will be extended to the higher-dimensional problems by a Kronecker product. In this way, the shape functions relative to this approach can be easily calculated with reasonably both computer space and time. The comparison between the obtained results and with available ones in the literature shows that the Mesh-free Spectral with Newmark Method (MS-NM) approach is a suitable and competitive approach for resolving time-space differential equations. [ABSTRACT FROM AUTHOR]

The paper is a continuation of the paper where the so-called coordinate and spectral equations were derived for finding the edge Green's functions on a branched surface having branch points of order two. The coefficients of those equations contain unknown constants. To find these constants, it is necessary to state restrictions for the solutions of the equations. After that, finding the unknown constants becomes possible, for example, by a numerical procedure of determining zeros of discrepancies. The paper is devoted to the statement of the problem of finding the unknown constants. As an example, the problem of scattering by two perpendicular half-lines is considered. As the result of using a rather subtle property of the spectral equation (symmetry associated with the reciprocity theorem), one can give a set of restrictions, in which the number of unknowns is equal to the number of restrictions. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a definition of very weak solutions to the homogenous Dirac-harmonic equations for differential forms. In this setting, applying the Gehring lemma and interpolation theorems, we establish a higher integrability of the Dirac operator based on the very weak solutions and explore the relation between weak solutions and very weak solutions. [ABSTRACT FROM AUTHOR]

As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse problem. The correct (vector) order can be found utilizing the available data. In this paper we offer an new method of solution of this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution U ^ (t , ξ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{U}(t, \xi)$$\end{document} evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of a system of governing pseudo-differential equations. [ABSTRACT FROM AUTHOR]

This paper is a brief review of results in the theory of covering mappings of metric spaces and vector metric spaces and its applications to implicit differential equations. For the Cauchy problem and boundary-value problems, we obtain existence conditions, estimates of solutions, and conditions of the continuous dependence of solutions on the parameters of the equation and initial and boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we will extend the strict stability to impulsive differential equations. By using Lyapunov functions, we will get some criteria for the strict stability of impulsive differential equations, and we can see that impulses do contribute to the system's strict stability behavior. An example is also given in this paper to illustrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

In the present paper, we carry out a systematic study of the flavor invariants and their renormalization-group equations (RGEs) in the leptonic sector with three generations of charged leptons and massive Majorana neutrinos. First, following the approach of the Hilbert series from the invariant theory, we show that there are 34 basic flavor invariants in the generating set, among which 19 invariants are CP-even and the others are CP-odd. Any flavor invariants can be expressed as the polynomials of those 34 basic invariants in the generating set. Second, we explicitly construct all the basic invariants and derive their RGEs, which form a closed system of differential equations as they should. The numerical solutions to the RGEs of the basic flavor invariants have also been found. Furthermore, we demonstrate how to extract physical observables from the basic invariants. Our study is helpful for understanding the algebraic structure of flavor invariants in the leptonic sector, and also provides a novel way to explore leptonic flavor structures. [ABSTRACT FROM AUTHOR]

This paper deals with the subject of infinitesimal variations of Euclidean submanifolds with arbitrary dimension and codimension. The main goal is to establish a Fundamental theorem for these geometric objects. Similar to the theory of isometric immersions in Euclidean space, we prove that a system of three equations for a certain pair of tensors are the integrability conditions for the differential equation that determines the infinitesimal variations. In addition, we give some rigidity results when the submanifold is intrinsically a Riemannian product of manifolds. [ABSTRACT FROM AUTHOR]

We consider an initial–boundary value problem for a nonlinear Sobolev-type second-order differential equation with a noncoercive source describing the behavior of charges in a semiconductor plasma in an external field. The local solvability of this problem is proved, and upper bounds are obtained for the solution blow-up time. [ABSTRACT FROM AUTHOR]

In this paper, we study a loaded modified diffusion equation (the Hallaire equation with the fractional derivative with respect to time). The compact finite difference schemes of Crank-Nicolson type of higher order is developed for approximating the stated problem on uniform grids with the orders of accuracy O(h4 + τ2−α) and O (h4 + τ2). A priori estimates are obtained for solutions of differential and difference equations. Stability of the suggested schemes and also their convergence with the rate equal to the order of the approximation error are proved. Proposed theoretical calculations are illustrated by numerical experiments on test problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with two logarithmic fourth order differential equations: the extended one-dimensional DLSS equation and its multi-dimensional analog. We show the global existence of solution in critical spaces, its convergence to equilibrium and the gain of spatial analyticity for these two equations in a unified way. [ABSTRACT FROM AUTHOR]

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with $f\in L^{m}(\Omega)$ ( $m\geqslant1$ ) and $\alpha(x)>0$ . Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$ , some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of $\alpha(x)$ . [ABSTRACT FROM AUTHOR]

A nonlocal problem with an integral matching condition is studied for a parabolic-hyperbolic equation with two lines of type change containing a nonlinear loaded term. The uniqueness of the solution of the problem is proved by the method of energy integrals and the existence, using the theory of integral equations. Classes and sufficient conditions are determined for given functions that ensure the unique solvability of the problem under study. [ABSTRACT FROM AUTHOR]

In this paper the Lie symmetry analysis is performed for the study of a generalized 2D Zakharov–Kuznetsov–Burgers equation with a variable dissipation. This equation originates from Zakharov–Kuznetsov equation supplemented with parabolic terms as in Burgers equation, which describes the nonlinear development of ion-acoustic waves in a magnetized plasma comprised of cold ions and hot isothermal electrons in the presence of a uniform magnetic field. We use Lie symmetry analysis to study this equation, which leads us reductions equations and solutions for this equation. We also get the low-order conservation laws of the equation that have been obtained using the corresponding symmetries of the family. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence of nonoscillatory solutions of higher-order neutral differential equations with distributed coefficients and delays. We use the $Banach$ contraction principle to obtain new sufficient condition for the existence of nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

In this paper, we examine the controllability problem of a distributed system governed by the two-dimensional Gurtin-Pipkin equation. We consider a system with compactly supported distributed control and show that if the memory kernel is a twice continuously differentiable function, such that its Laplace transformation has at least one root, then the system cannot be driven to equilibrium in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method to find the periodic solutions of differential equations. The key technique is to convert the problem of finding periodic solutions of differential equations into an optimization problem. Then by solving the corresponding optimization problem, we can find the periodic solutions of differential equations. Finally, some numerical results are presented to illustrate the utility of the technique. [ABSTRACT FROM AUTHOR]

Differential equations of fractional order appear in many applications in physics, chemistry and engineering. An effective and easy-to-use method for solving such equations is needed. In this paper, series solutions of the FDEs are presented using the homotopy analysis method (HAM). The HAM provides a convenient way of controlling the convergence region and rate of the series solution. It is confirmed that the HAM series solutions contain the Adomian decomposition method (ADM) solution as special cases. [ABSTRACT FROM AUTHOR]

This paper proposes a new effective pseudo-spectral approximation to solve the Sylvester and Lyapunov matrix differential equations. The properties of the Chebyshev basis operational matrix of derivative are applied to convert the main equation to the matrix equations. Afterwards, an iterative algorithm is examined for solving the obtained equations. Also, the error analysis of the propounded method is presented, which reveals the spectral rate of convergence. To illustrate the effectiveness of the proposed framework, several numerical examples are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper was to investigate the dynamics of the option pricing in the market through the two-dimensional time fractional-order Black–Scholes equation for a European put option. The Liouville–Caputo derivative was used to improve the ordinary Black–Scholes equation. The analytic solution is a powerful tool for describing the behavior of the option price in the European style market. In this study, analytic solution is carried out by the Laplace homotopy perturbation method. Moreover, the obtained solution showed that the Laplace homotopy perturbation method was an efficient method for finding an analytic solution of two-dimensional fractional-order differential equation. [ABSTRACT FROM AUTHOR]

In this paper we study subordination principles for fractional differential equations of Sobolev type in Banach space. With the help of the theory of Sobolev type resolvent families (known also as propagation family) as well as these subordination principles, we obtain the existence of mild solutions for this kind of equations. We study simultaneously the case 0 < α < 1 and 1 < α < 2 for the Caputo and Riemann-Liouville fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of the geometry of spacelike surfaces in de Sitter 5-space. We define invariants of the second fundamental form and consider their geometrical properties. We also investigate generic properties of the surfaces defined as solutions of the equations of asymptotic directions (AD) and lightlike binormal directions (BD) of spacelike surfaces. [ABSTRACT FROM AUTHOR]

This is the first paper of a conceived series under the common title “The boundary control method in inverse problems.” The aim of the series is to expound systematically an approach to inverse problems based upon its relationship with control theory. The 1d-variant of the method is shown with the example of the classical problem of recovering the density of an inhomogeneous string, and both dynamical and spectral statements of the problem are considered. The paper is written in such a way as to serve as an introduction to the multidimensional BC-method: the basic tools and constructions are amenable to further generalization to multidimensional problems. Bibliography: 31 titles. [ABSTRACT FROM AUTHOR]

In this paper we study the numerical method for a time-fractional Black–Scholes equation, which is used for option pricing. The solution of the fractional-order differential equation may be singular near certain domain boundaries, which leads to numerical difficulty. In order to capture the singular phenomena, a numerical method based on an adaptive moving mesh is developed. A finite difference method is used to discretize the time-fractional Black–Scholes equation and error analysis for the discretization scheme is derived. Then, an adaptive moving mesh based on an a priori error analysis is established by equidistributing monitor function. Numerical experiments support these theoretical results. [ABSTRACT FROM AUTHOR]

A system of equations that describe the static equilibrium of plasma in a magnetic field with three-dimensional magnetic surfaces of toroidal topology is obtained. The approach is based on the mixed representation for the magnetic field; the representation is similar to that used in the derivation of the classical Grad–Shafranov equation but modified with allowance for the possible axial asymmetry of magnetic surfaces. The system consists of three differential equations for three scalar functions; two of the equations can be written in the form of magnetic differential equations and the third one (the balance of forces in the direction perpendicular to the surface) serves as an analogue of the Grad–Shafranov equation. The obtained equations allow a simple transition to the limiting case of axial symmetry. An example of a solution of the Hill vortex type is presented under the assumption of weak axial asymmetry of the problem. The paper is dedicated to the memory of the brilliant physicist Vitaly D. Shafranov, who could think simply and very precisely. [ABSTRACT FROM AUTHOR]

We consider the problem of dynamic reconstruction of unknown boundary disturbances in a parabolic equation. The disturbances are assumed to occur in the Neumann boundary condition. Based on the Osipov-Kryazhimskii dynamic regularization method, the problem is reduced to that of constructing a feedback control law for some auxiliary differential equation. We indicate an information-noise- and roundoff-error-robust algorithm for solving this problem. [ABSTRACT FROM AUTHOR]

This paper proposes a new collocation method for estimating parameters of a partial differential equation (PDE), which uses Gaussian process (GP) as a basis function and is termed as Gaussian process for partial differential equation (GPPDE). The conventional method of estimating parameters of a differential equation is to minimize the error between observations and their estimates. The estimates are produced from the forward solution (numerical or analytical) of the differential equation. The conventional approach requires initial and boundary conditions, and discretization of differential equations if the forward solution is obtained numerically. The proposed method requires fitting a GP regression model to the observations of the state variable, then obtaining derivatives of the state variable using the property that derivative of a GP is also a GP, and finally adjusting the PDE parameters so that the GP derived partial derivatives satisfy the PDE. The method does not require initial and boundary conditions, however if these conditions are available (exactly or with measurement errors), they can be easily incorporated. The GPPDE method is evaluated by applying it on the diffusion and the Richards equations. The results suggest that GPPDE can correctly estimate parameters of the two equations. For the Richards equation, GPPDE performs well in the presence of noise. A comparison of GPPDE with HYDRUS-1D software showed that their performances are comparable, though GPPDE has significant advantages in terms of computational time. GPPDE could be an effective alternative to conventional approaches for finding parameters of high-dimensional PDEs where large datasets are available. [ABSTRACT FROM AUTHOR]

In this paper, the problem of robust stability for linear time-varying implicit dynamic equations is generally studied. We consider the effect of uncertain structured perturbations on all coefficient matrices of equations. A formula of stability radius with respect to dynamic structured perturbations acting on the right-hand side coefficients is obtained. In case where structured perturbations affect on both derivative and the right-hand side, the lower bounds for the stability radius are derived. The results are novel and extend many previous results about robust stability for time-varying ordinary differential/difference equations, time-varying differential algebraic equations and time-varying implicit difference equations. [ABSTRACT FROM AUTHOR]

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study persistence properties for a generalized Camassa-Holm equation with cubic nonlinearity, and we prove the persistence properties in weighted spaces of the solution to the equation, provided that the initial potential satisfies a certain sign condition. Our results extend the work of Brandolese (Int. Math. Res. Not. 22:5161-5181, ) on persistence properties to the Fokas-Olver-Rosenau-Qiao equation. In contrast to the Camassa-Holm equation with quadratic nonlinearity, the effect of cubic nonlinearity of the Fokas-Olver-Rosenau-Qiao equation on the persistence properties is rather delicate. [ABSTRACT FROM AUTHOR]

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems. [ABSTRACT FROM AUTHOR]