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Register a new userHow Scaling Symmetry Solves a Second-Order Differential Equation
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While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by quadrature (Lies Theorem on the solvability of differential equations). We illustrate this theorem by applying scale invariance to the equations for the hydrostatic equilibrium of stars in local thermodynamic equilibrium: Scaling symmetry reduces the Lane-Emden equation to a first-order equation between scale invariants un; vn, whose phase diagram encapsulates all the properties of index-n polytropes. From this reduced equation, we obtain the regular (Emden) solutions and demonstrate graphically how they transform under scale transformations.
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In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Greens function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Greens function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Greens function solution is numerically justified for the sinh-Gordon and Liouville equations. We also list two open problems leading to a more thorough characterizations of non-linearities admitting the obtained representation for the nonlinear Greens function.
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. If the solution is a special function that is studied in the literature, then the result is more manageable using the properties of that function. It is straightforward to find the regular and irregular singular points of such an equation by a computer algebra system. However, one needs the corresponding indices for a full analysis of the singularity structure. It is shown that the $theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure which can be visualized through a Riemann P-symbol leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying some applications belonging to different mathematical contexts.
The Greens function method which has been originally proposed for linear systems has several extensions to the case of nonlinear equations. A recent extension has been proposed to deal with certain applications in quantum field theory. The general solution of second order nonlinear differential equations is represented in terms of a so-called short time expansion. The first term of the expansion has been shown to be an efficient approximation of the solution for small values of the state variable. The proceeding terms contribute to the error correction. This paper is devoted to extension of the short time expansion solution to non-linearities depending on the first derivative of the unknown function. Under a proper assumption on the nonlinear term, a general representation for Greens function is derived. It is also shown how the knowledge of nonlinear Greens function can be used to study the spectrum of the nonlinear operator. Particular cases and their numerical analysis support the advantage of the method. The technique we discuss grants to obtain a closed form analytic solution for non-homogeneous non-linear PDEs so far amenable just to numerical solutions. This opens up the possibility of several applications in physics and engineering.
We show that all results of Yasar and Ozer [Comput. Math. Appl. 59 (2010), 3203-3210] on symmetries and conservation laws of a nonconservative Fokker-Planck equation can be easily derived from results existing in the literature by means of using the fact that this equation is reduced to the linear heat equation by a simple point transformation. Moreover nonclassical symmetries and local and potential conservation laws of the equation under consideration are exhaustively described in the same way as well as infinite series of potential symmetry algebras of arbitrary potential orders are constructed.
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