Partial differential equations admitting a given nonclassical point symmetry

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

本文确定了一类复合方程的古典对称分类和非古典对称分类。首先，基于微分特征列集算法确定了复合方程的古典对称分类。其次，确定了复合方程的非古典对称的分类。第一步，添加不变曲面条件与原方程组成一个新的偏微分方程组(PDEs)，利用符号计算软件Mathematica确定上面PDEs的对称对应的确定方程组(DTEs)；第二步，根据所得的DTEs进行非古典对称分类，得到复合方程中参数F(u)的具体形式。第三步，确定了非古典对称所对应的不变解以及精确解。所得的不变解和精确解无法利用古典对称得到，所以丰富了复合方程的精确解。 In this paper, the classifications of classical and nonclassical symmetries to a composite type equation are determined. Firstly, the classification of classical symmetries to the composite equation is determined based on the differential characteristic set algorithm. Secondly, the classification of nonclassical symmetries for the composite equation is determined. First step, adding invariant surface condition and the original equation composed a new system of partial differential equations (PDEs), and the determining equations (DTEs) of symmetry to PDEs are determined by using the symbolic computation software Mathematica. Second step, the nonclassical symmetries are classified by calculating DTEs, so we can obtain the specific form of F(u) which is the parameter of the composite equation. Third step, the invariant solutions and exact solutions of the corresponding nonclassical symmetry are determined. The invariant solutions and exact solutions cannot be obtained by classical symmetry, so enrich the exact solutions of the composite equation.

We consider nonclassical symmetries of partial differential equations (PDEs) in dimensions. Given a th‐order ordinary differential equation in the unknown we are able to find the most general scalar PDE of a given order which can be reduced via a nonclassical symmetry to .

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?

usual boundary conditions, and constraints u~U(t, z)~Em, where the control space U = U(t, z) is a given subset of Era. Here z =(z 1 . . . . . : ) are state variables and u =(u 1 .. . . , u m) control variables. As stated by P. K. RASHEVSKY [6], very general partial differential equations and systems can be written locally in the form (2). For instance, second order partial differential equations such as z~,+z~s= 1 u) can be immediately written in the form f ( t , s, z 1, zlt, z~,

Assessing basic research : Some partial indicators of scientific progress in radio astronomy

PDE blending surfaces with C2 continuity

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.

The primary aim is to develop a new class of pseudo-differential operators involving Linear Canonical Transform (LCT) and some of its essential results in Schwartz space and tempered distribution space. The secondary aim is to find applications of the LCT while constructing new general partial differential equations (Heat and Wave) and derive their solutions. Further, we have discussed particular cases of these partial differential equations. Furthermore, we have also shown the resolution of the solution of these equations graphically.

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al . ( Basak et al . 1999 J. Math. Anal. Appl. 202 , 604–622) and Yuan et al . ( Yuan et al . 2003 Syst. Control Lett. 50 , 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa ( Ichikawa 1982 J. Math. Anal. Appl. 90 , 12–44) and Luo & Liu ( Luo & Liu 2008 Stoch. Proc. Appl. 118 , 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

In this paper, classical and nonclassical symmetries of the potential KdV equation are considered. A catalogue of symmetry reductions for potential KdV equation is obtained using the classical Lie method and nonclassical method due to Bluman and Cole.[1] The Lie algebra consists of five finite parameter Lie group transformations; two being the scaling symmetry and the others being translations. By using the nonclassical method, four finite parameter group transformations are obtained. Two different types of solutions, approximate and exact solutions, are found by using the symmetries. Using the classical symmetries, only approximate series solutions are constructed, but using the nonclassical symmetries, both the exact solution and the approximate solution were found. The advantage of nonclassical symmetries is that the exact solution of the KdV equation can be found with it. Perturbation method is also used for approximate solutions.

The partial differential equation that describes the flow, of non-Newtonian, power-law, slightly compressible fluids in porous media is derived. An approximate solution, in closed form, is developed for the unsteady-state flow behavior and verified by. two different methods. Using the unsteady-state solution, a method for analyzing injection test data is formulated and used to analyze four injection tests. Theoretical results were used to derive steady-state equations of flow, equivalent transient drainage radius, and a method for analyzing isochronal test data. The theoretical fundamentals of the flow, of non-Newtonian power-law fluids in porous media are established. Introduction Non-Newtonian power-law fluids are those that obey the relation = constant. Here, is the viscosity, e is the shear rate at which the viscosity is measured, and n is a constant. Examples of such fluids are polymers. This paper establishes the theoretical foundation of the flow of such fluids in porous media. The partial differential equation describing this flow is derived and solved for unsteady-state flow. In addition, a method for interpreting isochronal tests and an equation for calculating the equivalent transient drainage radius are presented. The unsteady-state flow solution provides a method for interpreting flow tests (such as injection tests).Non-Newtonian power-law fluids are injected into the porous media for mobility control, necessitating a basic porous media for mobility control, necessitating a basic understanding of the flow behavior of such fluids in porous media. Several authors have studied the porous media. Several authors have studied the rheological properties of these fluids using linear flow experiments and standard viscometers. Van Poollen and Jargon presented a theoretical study of these fluids. They described the flow by the partial differential equation used for Newtonian fluids and accounted for the effect of shear rate on viscosity by varying the viscosity as a function of space. They solved the equation numerically using finite difference. The numerical results showed that the pressure behavior vs time differed from that for Newtonian fluids. However, no methods for analyzing flow-test data (such as injection tests) were offered. This probably was because of the lack of analytic solution normally required to understand the relationship among the variables.Recently, injectivity tests were conducted using a polysaccharide polymer (biopolymer). The data showed polysaccharide polymer (biopolymer). The data showed anomalies when analyzed using methods derived for Newtonian fluids. Some of these anomalies appeared to be fractures. However, when the methods of analysis developed here were applied, the anomalies disappeared. Field data for four injectivity tests are reported and used to illustrate our analysis methods. Theoretical Consideration General Consideration The partial differential equation describing the flow of a non-Newtonian, slightly compressible power-law fluid in porous media derived in Appendix A is ..........(1) where the symbols are defined in the nomenclature. JPT P. 155

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

Potential nonclassical symmetries and solutions of fast diffusion equation