Improved Error Bounds for Approximations of High-Frequency Wave Propagation in Nonlinear Dispersive Media

Solving systems of partial differential equations (linear or nonlinear) with dirchelet boundary conditions has rarely made use of the Adomian decompositional method. The aim of this paper is to obtain the exact solution of some systems of linear and nonlinear partial differential equations using the adomian decomposition method.After having generated the basic principles of the general theory of this method, five systems of equations are solved, after calculation of the algorithm.Our results suggest that the use of the adomian method to solve systems of partial differential equations is efficient.However, further research should study other systems of linear or nonlinear partial differential equations to better understand the problem of uniqueness of solutions and boundary conditions.

The method of Lie group transformation is used to obtain an approximate analytical solution to the system of first-order quasilinear partial differential equations that govern a one-dimensional unsteady planer, cylindrically symmetric and spherically symmetric motion in a non-ideal gas, involving strong shock waves. Invariance groups admitted by the governing system of partial differential equations, which are indeed continuous group of transformations under which the system of partial differential equations remains invariant, are determined, and the complete Lie algebra of infinitesimal symmetries is established. The infinitesimal generators are used to construct the similarity variables. These similarity variables are used to reduce the governing system of partial differential equations into a system of ordinary differential equations.

An explicit formula for the mathematical expectation and second moment functions of a solution to a linear system of ordinary differential equations with a random parameter and a vector random righthand side is obtained. The problem is reduced to the deterministic Cauchy problem for systems of first-order linear partial differential equations. We obtain an explicit formula for a solution of linear systems of partial differential equations of the first order with constant coeffcients. An example is given showing that random factors can have a stabilizing effect on a linear system of differential equations.

The linear systems of partial differential equations of the first order with the identical main partsis considered. Аpplying the well-known relation between a normal system of ordinary differentialequations and a linear system of partial differential equations of the first order with the samemain parts, the existence of integral basis of a linear inhomogeneous system of partial differentialequations of the first order adjoining to some solution of the same linear inhomogeneous systemof differential equations with partial derivatives of the first order is proved. A sign at which thenonlinear system of ordinary differential equations has a neighborhood such that any solution withinitial values from it tends to zero is found. Using the equivalence of a linear system of partialdifferential equations of the first order with identical main parts to a linear differential equationwith partial derivatives of the first order, the existence of integral basis of the adjoining to zerolinear homogeneous system of partial differential equations of first order with nonlinear coefficientsis shown.

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.

Pollen indicators of human activity

This article reports an unsteady two-dimensional Magneto-hydrodynamic (MHD) boundary layer flow of an incompressible electrically conducting fluid over a slippery stretching sheet surrounded in a porous medium. The Roseland boundary layer approximation with the radiative heat flux is employed within the current analysis. The influence of the velocity slip, thermal radiation, heat source, and buoyancy force is also considered within the current analysis, which makes significant effects on the flow field passages. The unsteady system of non-dimensional partial differential equations (PDEs) with corresponding boundary conditions are solved by implementing the explicit finite difference scheme. In the presence of pertinent parameters such as viscous dissipation, heat source or sink, Prandtl number, Grashof number, thermal radiation, magnetic field, and Darcy number, the accurate movement of the electrically conducting fluid over a slippery sheet is shown graphically in the form of velocity, temperature, skin friction coefficient, and Nusselt number. Unlike the other studies, wherein the system of PDEs is commonly transformed into a system of ordinary differential equations via the similarity transformations, the current study provides an efficient numerical procedure to solve a given system of PDEs without using the similarity transformations which exemplify the precise movement of an electrically conducting fluid over a slippery surface. It has been anticipated that the current boundary layer analysis would provide a platform for solving the system of the nonlinear PDEs of the other unsolved boundary layer models that are associated with the two-dimensional unsteady MHD flow over a slippery stretching surface embedded in a porous medium.

The theory of analytic systems of partial differential equations was first systematically investigated by Riquier and Elie Cartan around 1900. The existence of local solutions involves an algebraic problem, finding formal power series solutions, and an analytic problem, proving the convergence of formal power series solutions. Cartan defined the notion of an involutive system of partial differential equations and, using his theory of exterior differential systems, was able to show the existence of formal power series solutions for involutive partial differential equations of first order and to prove the convergence by the Cauchy-Kowalewski theorem. His result was extended by Kihler to systems of partial differential equations of higher order and is known today as the Cartan-Kihler theorem. Adjoining to a system of partial differential equations of order k the equations obtained by differentiating the original equations gives rise to a system of partial differential equations of order k +1, the prolongation of the system, which has the same solutions as the original equations. Cartan conjectured that, by prolonging a system a sufficient number of times, one would obtain an involutive system; this result was proved by Kuranishi in 1957 within the framework of Cartan's theory of exterior differential systems, and is now referred to as the Cartan-Kuranishi prolongation theorem. In 1961, Spencer introduced, in his fundamental paper [6] on the deformation of pseudogroup structures, certain cohomology groups Hkj associated to a partial differential equation (see ? 3), which are dual to homology groups of a Koszul complex; and so the cohomology groups Hki vanish for all sufficiently large k (see Lemma 3.1). The vanishing of these cohomology groups was shown by Serre to be equivalent to Cartan's notion of involutiveness (see V. W. Guillemin and S. Sternberg [3]). It then became possible to analyse the role played by involutiveness in Cartan's theory of partial differential equations. In this paper, we prove the Cartan-Kahler theorem for systems of linear partial differential equations formulated in terms of Ehresmann's theory of

94 - Method of Characteristics

In this paper, we study the possibility of the existence of a universal solution of the Cauchy problem for the partial differential equation (PDE) systems in the case if this system is overdetermined so that the new overdetermined system of PDE contains all solutions of the initial PDE system and, in addition, reduces to the ordinary differential equation (ODE) systems, whose solution is then found. To do this, the article discusses the modification of the method of finding particular solutions for any overdetermined systems of differential equations by reduction to overdetermined systems of implicit equations. In the previous papers of the authors, a method was proposed for finding particular solutions for overdetermined PDE systems. In this method, in order to find solutions it is necessary to solve systems of ordinary implicit equations. In this case, it can be shown that the solutions that we need cannot depend on a continuous parameter, i.e. they are no more than countable. In advance, there is a need for such an overriding of the systems of differential equations, so that their general solutions are no more than countable. Such an initial overdetermination is rather difficult to achieve. However, the proposed method also allows to reduce the overdetermined systems of differential equations not only up to systems of implicit equations, but also up to the PDE systems of dimension less than that of the initial systems of PDE. In particular, under certain conditions, reduction to the ODE systems is possible. It is proposed to choose solutions for the overdetermined PDE systems using the parameterized Cauchy problem, which is posed for parameterized ODE systems under certain conditions. The solution of this Cauchy problem is some function of the initial data and their derivatives. In order to find the solution of any corresponding Cauchy problem for the initial system of PDE, it is sufficient to calculate the universal solver for the reduced ODE system once. In this case, the solution will not only exist and be unique, but will also depend continuously on the initial data, since this holds for ODE systems. The purpose of this paper is to study the Cauchy problem with the possibility of its universalization and the parameterized Cauchy problem as a whole for arbitrary PDE systems.

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

Many works are devoted to control theory but most of them are related to ordinary differential equations. Of the partial differential equations, mathematical physics equations are most often considered, for example, wave equations. In the article by Makarov O.A. "Controllability of an evolutionary system of partial differential equations. Visnyk of V. N. Karazin Kharkiv National University, Series "Mathematics, Applied Mathematics and Mechanics", 2016, Vol. 83. pp. 47-56" a system of partial differential equations has previously been considered. The complete controllability of such systems in the L. Schwartz space under certain control conditions has been investigated in this work. In particular, it was proved that if the eigenvalues of the system matrix are real, then there is a time-independent control. In addition, the case when the eigenvalues are imaginary was investigated. The purpose of this article is to study the controllability of a system of linear partial differential equations for an arbitrary matrix of the system under the constraints on the search for control in the form $u(x,t)=u(x)\cdot\exp(-\alpha t)$, where the vector of the function $u(x)$ belongs to the L. Schwartz space. Necessary and sufficient conditions for the complete controllability of this system are obtained and examples of both controllable and uncontrollable systems are given. As a consequence of this theorem, sufficient conditions for the complete controllability of the system are obtained, the real parts of the eigenvalues of the matrix $P(s)$ are bounded from above or below. In addition, it is proved that if the spatial variable belongs to the space $\mathbb{R}$, then such a system is completely controllable. Examples are given for each case. A partial differential equation of the second order in time is also considered. It has been proven that if the roots of the characteristic equation satisfy the conditions of the criterion, then this equation is completely controllable. Thus, the Helmholtz equation are completely controllable.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

Some systems of differential equations with partial derivatives are studied by using the properties of Gâteaux differentiable functions on commutative algebras. The connection between solutions of systems of partial differential equations and components of monogenic functions on the corresponding commutative algebras is shown. We also give some examples of systems of partial differential equations and find their solutions.

The method of lines used in conjunction with a sophisticated ordinary-differential-equations integrator is an effective approach for solving nonlinear, partial differential equations and is applicable to the equations describing fluid flow through porous media. Given the initial values, the integrator is self-starting. Subsequently, it automatically and reliably selects the time step and order, solves the nonlinear equations (checking for convergence, etc.), and maintains a user-specified time-integration accuracy, while attempting to complete the problems in a minimal amount of computer time. The advantages of this approach, such as stability, accuracy, reliability, and flexibility, are discussed. The method is applied to reservoir simulation, including high-rate and gas-percolation problems, and appears to be readily applicable to problems, and appears to be readily applicable to compositional models. Introduction The numerical solution of nonlinear, partial differential equations is usually a complicated and lengthy problem-dependent process. Generally, the solution of slightly different types of partial differential equations requires an entirely different computer program. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and highly reliable computer programs for automatically solving complicated systems of ordinary differential equations have become available. These computer programs feature variable-order methods and automatic time-step and error control, and are capable of solving broad classes of ordinary differential equations. This paper discusses how these sophisticated ordinary-differential-equation integrators may be used to solve systems of nonlinear partial differential equations. partial differential equations.The basis for the technique is the method of lines. Given a system of time-dependent partial differential equations, the spatial variable(s) are discretized in some manner. This procedure yields an approximating system of ordinary differential equations that can be numerically integrated with one of the recently developed, robust ordinary-differential-equation integrators to obtain numerical approximations to the solution of the original partial differential equations. This approach is not new, but the advent of robust ordinary-differential-equation integrators has made the numerical method of lines a practical and efficient method of solving many difficult systems of partial differential equations. The approach can be viewed as a variable order in time, fixed order in space technique. Certain aspects of this approach are discussed and advantages over more conventional methods are indicated. Use of ordinary-differential-equation integrators for simplifying the heretofore rather complicated procedures for accurate numerical integration of systems of nonlinear, partial differential equations is described. This approach is capable of eliminating much of the duplicate programming effort usually associated with changing equations, boundary conditions, or discretization techniques. The approach can be used for reservoir simulation, and it appears that a compositional reservoir simulator can be developed with relative ease using this approach. In particular, it should be possible to add components to or delete components possible to add components to or delete components from the compositional code with only minor modifications. SPEJ P. 255