Optimal Gas-Power Flow Calculation With Lightweight Gas Flow Dynamics

Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement.

The paper considers a mixed initial-boundary value problem for a parabolic equation with nonhomogeneous boundary conditions. The classical approach to search for analytical solution of such problems in the first phase involves variable substitution, leading to a problem with homogeneous boundary conditions. Reference materials [1] give, as a rule, the simplest types of variable substitutions where new and old unknown functions differ by a term, linear in the spatial variable. The form of this additive term depends on the type of the boundary conditions, but is in no way related to the equation under consideration. Moreover, in the case of the second boundary-value problem, it is necessary to use a quadratic additive, since a linear substitution for this type of conditions may be unavailable. The courseware [2] - [4], usually, ends only with the first boundary-value problem generally formulated. The paper considers a substitution that takes into account, in principle, the form of a linear differential operator. Namely, as an additive term, it is proposed to use the parametrically time-dependent solution of the boundary value problem for an ordinary differential equation obtained from the original partial differential equation by the method of separation of the Fourier variables. The existence of the proposed substitution for boundary conditions of any type is proved by the example of a non-stationary heat-transfer equation with the heat exchange available with the surrounding medium. In this case, the additive term is a linear combination of hyperbolic functions. It is shown that, in addition to the insensitivity to the type of boundary conditions, the advantages of a new substitution in comparison with the traditional linear (or quadratic) one include a much simpler structure of the solution obtained. Just the described approach allows us to obtain a solution with a clearly distinguished stationary component, in case a stationarity occurs, for values of the time variable that exceeds relaxation time. This effect extends not only to parabolic equations, but is also observed in equations describing oscillatory processes with damping. It is also shown that in the simplest particular case of an equation with constant coefficients without heat exchange with the surrounding medium, the proposed substitution becomes appropriately linear and coincides with the classical one. The use of the proposed approach allows students majoring in engineering to expand a range of educational and research tasks while learning a course in partial differential equations.

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearity domain. To improve the generalizability, we introduce the novel approach of employing multi-task learning techniques, the uncertainty-weighting loss and the gradients surgery, in the context of learning PDE solutions. The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multiple related PDEs, which are obtainable by varying the PDE parameterization coefficients, to generalize better on the original PDE. Encouraging the network pay closer attention to the high nonlinearity domain regions that are more challenging to learn, we also propose adversarial training for generating supplementary high-loss samples, similarly distributed to the original training distribution. In the experiments, our proposed methods are found to be effective and reduce the error on the unseen data points as compared to the previous approaches in various PDE examples, including high-dimensional stochastic PDEs.

In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.

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.

Partial differential equation is an equation containing a partial derivative of one or more dependent variables on more than one independent variable. In the differential equation there are coefficients in the form of constants and functions. The solution of a partial differential equation whose coefficients are constants is easily determined. However, the solution of the differential equations whose coefficients are functions is quite difficult to determine. One method that can be used to determine the solution is by using Taylor polynomial. This method can be used in second-order linear partial differential equation with coefficient of function with two independent variables. The purpose of this research is to determine the Taylor polynomial solution on second-order linear partial differential equation. In this research we get solution from second-order linear partial differential equation by assuming solution in the form of polynomial of Taylor having degree 𝑁𝑁 𝑢𝑢 (𝑥𝑥, 𝑦𝑦) = 􀷍􀷍αα𝑟𝑟, 𝑠𝑠 (𝑥𝑥-𝑐𝑐0) 𝑟𝑟 (𝑦𝑦-𝑐𝑐1) 𝑠𝑠, 𝑁𝑁𝑠𝑠 = 0𝑁𝑁𝑟𝑟 = 0 with αα𝑟𝑟, 𝑠𝑠 = 1𝑟𝑟! 𝑠𝑠! 𝑢𝑢 (𝑟𝑟, 𝑠𝑠) (𝑐𝑐0, 𝑐𝑐1) is the Taylor polynomial coefficient, or can be expressed in terms of the matrix equation 𝑢𝑢 (𝑥𝑥, 𝑦𝑦) = 𝑋𝑋𝑋𝑋𝑋𝑋

Cellular Neural Network (CNN) has been used for solving Partial Differential Equations (PDE). However, the equivalence and stability of system should be considered carefully in a particular problem. In this paper, we introduce the model CNN for solving set of two PDEs describing water flow channels (called Saint Venant equation). We analyze the approximation and topological equivalence issues between Cellular Partial Difference Differential Equation (CPDDE) and its original PDEs. The stability of CNN system is also proved from discovering the equilibrium of the state and output of each cell. The paper has 4 parts. After introduction, part 2 gives a two-layered CNN 1D model for solving PDE Saint Venant equation. In the part 3 the equivalence and stability of the CNN model are proved, then simulation using FPGA. The conclusions are given in the last part.

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis. Sophus Lie (1842–1899) determined all continuous transformation groups of the 2D plane and gave normal forms for any ordinary differential equation that is invariant under one of those groups. I deal with the extension of Lie's program to second-order PDEs in z(x, y). The starting point to this endeavour is a previously unknown paper by Amaldi from 1901, which claims to have completed Lie's classification of groups acting in (x, y, z)-space. I also present a Maple procedure (‘LHSO1_PDE_Solver’) for solving systems of linear, homogeneous first-order PDEs that performs better on this class than Maple's built-in PDE system solver.

Simplified non-linear dynamical equations of circular cylindrical shells are obtained on the basis of asymptotic analysis. The original partial differential equations (PDEs), which have eighth order in the longitudinal coordinate, are replaced by two sets of equations that have fourth order on this coordinate. The first set contains non-linear and dynamic PDEs. They describe the oscillations in the inner region of the shell. For small non-linearities, they coincide with the well-known linear semi-membrane dynamical equations. Ordinary differential equations of the edge effect allow one to satisfy all the boundary conditions. They are linear and quasistatic. An important question about the boundary conditions for the simplified equations is considered. We also discuss the solution of the obtained boundary value problems.

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