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Register a new userSymmetry Analysis for a Fourth-order Noise-reduction Partial Differential Equation
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Andronikos Paliathanasis
Publication date
2020
fields
Physics
and research's language is
English
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We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.
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We study a one-parameter family of the fourth-order ordinary differential equations obtained by similarity reduction of the modifed Sawada-Kotera equation. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four. We make this polynomial Hamiltonian from the viewpoint of accessible singularity and local index. We also give its symmetry and holomorphy conditions. These properties are new. Moreover, we introduce a symmetric form in dimension five for this Hamiltonian system by taking the two invariant divisors as the dependent variables. Thanks to the symmetric form, we show that this system admits the affine Weyl group symmetry of type $A_2^{(2)}$ as the group of its B{a}cklund transformations.
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.
We discuss some open problems and recent progress related to the 4th order Paneitz operator and Q curvature in dimensions other than 4.
This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L{e}vy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Lo`{e}ve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.
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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