Equations Of Kind Research Articles

Application of an Ansatz Method on a Delay Model With a Proportional Delay Parameter

Application of an Ansatz Method on a Delay Model With a Proportional Delay Parameter

Smooth criteria for two kinds of equations in pressure

Smooth criteria for two kinds of equations in pressure

Constructing artificial boundary condition of dispersive wave systems by deep learning neural network

To solve one dimensional dispersive wave systems in an unbounded domain, a uniform way to establish localized artificial boundary conditions is proposed. The idea is replacing the half-infinite interval outside the region of interest with a super element which exhibits the same dynamics response. Instead of designing the detailed mechanical structures of the super element, we directly reconstruct its stiffness, mass, and damping matrices by matching its frequency-domain reaction force with the expected one. An artificial neural network architecture is thus specifically tailored for this purpose. It comprises a deep learning part to predict the response of generalized degrees of freedom under different excitation frequencies, along with a simple linear part for computing the external force vectors. The trainable weight matrices of the linear layers correspond to the stiffness, mass, and damping matrices we need for the artificial boundary condition. The training data consists of input frequencies and the corresponding expected frequency domain external force vectors, which can be readily obtained through theoretical means. In order to achieve a good result, the neural network is initialized based on an optimized spring-damper-mass system. The adaptive moment estimation algorithm is then employed to train the parameters of the network. Different kinds of equations are solved as numerical examples. The results show that deep learning neural networks can find some unexpected optimal stiffness/damper/mass matrices of the super element. By just introducing a few additional degrees of freedom to the original truncated system, the localized artificial boundary condition works surprisingly well.

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

Generalized multiple scale approach to the problem of a taut string traveled by a single force

The strongly nonlinear dynamics of taut strings, traveled by a force moving with uniform velocity, is analyzed. A change of variable is performed, which recasts the equations of motion in terms of a linearized dynamic displacement, measured from the nonlinear quasi-static response. Under the hypothesis the load velocity is far enough from the celerity of the string, the system appears in the form of linear PDEs whose coefficients are slowly variable in time. Since the classic perturbation methods are not applicable to such kind of equations, the Generalized Method of Multiple Scales is developed, by directly attacking the PDEs, to derive asymptotic solutions. The validity of the analytical predictions is assessed by comparisons with numerical simulations, aimed to prove the accuracy of (1) linearization, and (2) the asymptotic approach.

Comprehensive review of numerical schemes based on Hermite wavelets

Differential and integral equations are encountered in many applications of science and engineering and many mathematical models have also been formulated in terms of these equations. Due to some shortcomings of the already existing numerical methods, researchers are making efforts to find more efficient alternatives for obtaining solutions of many practical and physical problems giving rise to differential or integral equations. As a result, wavelet methods have found their way for the numerical solution of the resulting different kinds of equations. So this review paper intends to provide the great utility, accuracy and employability of Hermite wavelets to address situations of various areas of applied mathematics, physics, biology, optimal control systems, communication theory, queuing theory, medicine and many other scientific and engineering problems.

On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral

Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.

On the Double ARA-Sumudu Transform and Its Applications

The main purpose of this work is to present a new double transform called the double ARA-Sumudu transform (DARA-ST). The application of the new double transform to some basic functions and the master properties are introduced. The convolution and existence theorems are also presented and proved. These new results are implemented to obtain the solution of partial differential equations (PDEs), integral equations (IEs) and functional equations. We obtain new formulas for solving families of PDEs. The latter ones are used to obtain exact solutions of some familiar PDEs such as the telegraph equation, the advection–diffusion equation, the Klein–Gordon equation and others. Moreover, a simple formula for solving a special kind of integral equations is presented and implemented in some applications. The outcomes show that DARA-ST is useful and efficient in handling such kinds of equations.

On Fractional Hybrid Non-Linear Differential Equations Involving Three Mixed Fractional Orders with Boundary Conditions

In this paper, we study a class of non-linear fractional hybrid differential equations involving three mixed fractional orders with boundary conditions. Under weak assumptions, a formula of solutions is constructed and the existence results of the solutions for the problem are established. The results can be used to solve more general fractional hybrid equations, such as the general variable coefficient fractional hybrid Langevin equations. Moreover, the form of the solution for this kind of equation can provide a theoretical basis for the further study of the positive solution and its symmetry. We provide an example to support our main result.

The discrete Burgers equation

In this paper I give a short review on the continuous, semi-discrete and discrete Burgers equations, which are featured as integrable equations that are linearisable. The review focuses more on connections of these three kinds of equations and connections with (2+1)-dimensional systems. The continuous Burgers hierarchy is a reduction of the derivative nonlinear Schrödinger hierarchy. The later is a result of the squared eigenfunction symmetry constraint of the modified Kadomtsev–Petviashvili system. The semi-discrete Burgers hierarchy can be considered as Bäcklund transformations of the continuous Burgers hierarchy. They are also reductions of the relativistic Toda hierarchy, which is a consequence of the squared eigenfunction symmetry constraint of the semi-discrete modified Kadomtsev–Petviashvili system. The fully discrete Burgers equation is the Bianchi identity of the Bäcklund transformation for the continuous Burgers hierarchy. It is 3-point lattice equation, three-dimensionally consistent, and linearisable.

Stability Results of Some Fractional Neutral Integrodifferential Equations with Delay

Differential equations with fractional derivative are being extensively used in the modelling of the transmission of many infective diseases like HIV, Ebola, and COVID-19. Analytical solutions are unreachable for a wide range of such kind of equations. Stability theory in the sense of Ulam is essential as it provides approximate analytical solutions. In this article, we utilize some fixed point theorem (FPT) to investigate the stability of fractional neutral integrodifferential equations with delay in the sense of Ulam-Hyers-Rassias (UHR). This work is a generalized version of recent interesting works. Finally, two examples are given to prove the applicability of our results.

Adaptive finite element method for two-dimensional time-harmonic magnetic induction intensity equations

In this paper, we propose, analyze, and numerically validate an adaptive finite element method for two-dimensional time-harmonic magnetic induction intensity equations as well as their Perfectly Matched-Layer (PML) equations. Based on Hodge decomposition, the equations are transformed into scalar elliptic boundary value problems and numerically solved by using the P1 finite element method. Also, we can solve another basic quantity E concerned in physics. A posterior error indicator based on a superconvergent functional value recovery is considered for the two kinds of equations. Numerical experiments are presented to illustrate the effectiveness of the a posterior error indicator and the corresponding adaptive algorithm.

ON NONTRIVIAL SOLVABILITY OF ONE CLASS OF NONLINEAR INTEGRAL EQUATIONS WITH CONSERVATIVE KERNEL ON THE POSITIVE SEMI-AXIS

The work is devoted to a special class of nonlinear integral equations on the positive semi-axis with conservative kernel that corresponds to a nonlinear operator, for which the property of complete continuity in the space of bounded functions fails. In different special cases this class of equations has applications in particular branches of mathematical physics. In particular, this kind of equations can be met in the radiative transfer theory, kinetic theory of gases, kinetic theory of plasma and in the $p$-adic open-closed string theory. Using a combination of special iterations with the monotonic operator theory methods, that work in defined conical segments it is possible to prove a constructive existence theorem of nonnegative nontrivial bounded solution that has finite limit at infinity. The asymptotics of the constructed solution will also be studied. It is also given an example of nonlinear equation, for which the uniqueness of the solution in the space of bounded functions fails. At the end of the paper will consider some classes of equations both applied and pure theoretical character.

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space ℝN. As an application, we then show the exact null-controllability for this semilinear heat equation in ℝN. The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in ℝN. This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.

Аналитическое решение первой задачи динамики манипуляторов

The aim is to solve the problem of deriving an explicit analytical form and the cumbersome of the equa-tions of dynamics of body systems. The research methods refer to the mechanics of body systems and systems analysis. The research results allow to write out formulas for calculating forces and moments of forces in the joints of the systems of bodies with one open branch. It is demonstrated in examples of writing out analytical types of equations of dynamics of industrial robot arms with three and six degrees of freedom in space. Three kinds of equations of dynamics were obtained for such manipula-tors. The first equations are written out in scalar-coordinate form with explicit quasi-accelerations and ve-locities, whose role is played by the projections of absolute angular accelerations and velocities of bodies on their connected axes. The second ones are written in vector-matrix form and are obtained from the former in the process of replacing quasi-accelerations by relative linear and angular accelerations of bodies with the allocation of a symmetric matrix of inertial coefficients. The third kind of equations of dynamics is ob-tained from the second one in the process of replacing quasi-velocities by relative linear and angular veloci-ties of bodies. In the third form, the centrifugal, Coriolis, and gyroscopic inertial forces are clearly ex-pressed. Gyroscopic inertial forces allow us to simplify the formula for calculating the power consumption of drives, as well as to simplify the Timofeev formula for calculating the driving forces and moments of forces that provide control of the program motion of manipulator bodies with a given quality. A technique for reusing formulas for manipulators with matching kinematic diagrams of their subsystems is demonstra-ted in the examples . Geometric, kinematic, static and inertial parameters of bodies are explicitly expressed in the equations of dynamics The multipliers for accelerations and products of velocities in the equations of dynamics are optimal in the sense of the minimum of arithmetic operations (additions and multiplica-tions) required for their calculations. Conclusion. All analytical types of equations of dynamics are veri-fied. They occupy several lines of text and further simplification is practically impossible.

An Algorithm Derivative-Free to Improve the Steffensen-Type Methods

Solving equations of the form H(x)=0 is one of the most faced problem in mathematics and in other science fields such as chemistry or physics. This kind of equations cannot be solved without the use of iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, are usually considered to solve these problems when H is a non-differentiable operator due to its accuracy and efficiency. However, in general, the accessibility of these iterative methods is small. The main interest of this paper is to improve the accessibility of Steffensen-type methods, this is the set of starting points that converge to the roots applying those methods. So, by means of using a predictor–corrector iterative process we can improve this accessibility. For this, we use a predictor iterative process, using symmetric divided differences, with good accessibility and then, as corrector method, we consider the Center-Steffensen method with quadratic convergence. In addition, the dynamical studies presented show, in an experimental way, that this iterative process also improves the region of accessibility of Steffensen-type methods. Moreover, we analyze the semilocal convergence of the predictor–corrector iterative process proposed in two cases: when H is differentiable and H is non-differentiable. Summing up, we present an effective alternative for Newton’s method to non-differentiable operators, where this method cannot be applied. The theoretical results are illustrated with numerical experiments.

The Cauchy problem for inhomogeneous parabolic Shilov equations

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

Multifeatured Image Retrieval Techniques Based on Partial Differential Equations for Online Shopping

In today’s rapid development of network and multimedia technology, the booming of electronic commerce, users in the network shopping species of images and other multimedia information showing geometric growth, in the face of this situation, how to find the images they need in the vast amount of online shopping images has become an urgent problem to solve. This paper is based on the partial differential equation to do the following research: Based on the partial differential equation is a kind of equation that simulates the human visual perception system to analyze images; based on the summary of the advantages and disadvantages of multifeature image retrieval technology, we propose a multifeature image retrieval technology method based on the partial differential equation to alleviate the indexing imbalance caused by the mismatch of multifeature image retrieval technology distribution. To improve the search speed of the data-dependent locally sensitive hashing algorithm, we propose a query pruning algorithm compatible with the proposed partial differential equation-based multifeature image retrieval technology method, which greatly improves the retrieval speed while ensuring the retrieval accuracy; to implement the data-dependent partial differential equation algorithm, we need to distribute the data set among different operation nodes, and to better achieve better parallelization of operations, we need to measure the similarity between categories, and we achieve the problem of distributing data among various categories in each operation node by introducing a clustering method with constraints. The purpose of this article for image recognition is for better shopping platforms for merchants. This algorithm has trained multiple samples and has data support. The experimental results show that our proposed data set allocation method shows significant advantages over the data set allocation method that does not consider category correlation. However, the image features used in image retrieval systems are often hundreds or even thousands of dimensions, and these features are not only high in dimensionality but also huge in number, which makes image retrieval systems encounter an inevitable problem—“dimensionality disaster.” To overcome this problem, scholars have proposed a series of approximate nearest neighbor methods, but multifeature image retrieval techniques based on partial differential equations are more widely used in people’s daily life.

A homotopy method for solving multilinear systems with strong completely positive tensors

A basic problem in both pure and applied mathematics is solving various kinds of equations. In recent years, multilinear systems of equations have received a great deal of attention and active research. However, most studies are focused on those multilinear systems with the tensors involved being strong M-tensors. It is well known that many tensors from real problems are non-negative, including strong completely positive tensors. In this paper, we consider the multilinear system of equations whose coefficient tensors are strong completely positive tensors of even order. First we show that this multilinear system has a unique real solution. Secondly, we propose a homotopy approximation method for solving the multilinear system, which has global convergence, and it possesses locally superlinear convergence under certain conditions. Preliminary numerical experiment results show that the proposed algorithm is effective.

New oscillatory results for non-linear delay dynamic equations with super-linear neutral term

A new approach is introduced for the oscillatory solutions of the second order non-linear delay dynamic equations on time scale with a super-linear neutral term. The result are obtained through various integral and differential relations. The Kamenev and Philos-type criteria for oscillation are also discussed. The results improve and extend the ones reported in the literature in the sense that the inequalities are easy to verify and results cover various kinds of equations by choosing different time scales. Examples for particular time scales are given for illustration.