Similarity Solution of the Flamant Problem by Means of a One-Parameter Group Transformation

The two-dimensional point load problem on a linear elastic half plane is solved using a one-parameter group transformation of the biharmonic equation. This solution technique allows the incorporation of the singular load directly through use of the Dirac delta function. The resulting fourth-order ordinary differential equation, while formidable in appearance, is exact and is readily solved through simple integration.     

Boundary-layer flow of a micropolar fluid due to a stretching wall

Summary  A Theoretical analysis is carried out to study the boundary-layer flow over a continuously moving surface through an otherwise quiescent micropolar fluid. The transformed boundary-layer equations are solved numerically for a power-law surface velocity using the Keller-box method. The effects of the micropolar K and exponent m parameters on the velocity and microrotation field as well as on the skin-friction group are discussed in a detailed manner. It is shown that there is a near-similarity solution of this problem. The accuracy of the present solution is also discussed. Accepted for publication 1 April 1996     

On the Self-Similar Jordan Arcs Admitting Structure Parametrization

We study the attractors γ of a finite system of contraction similarities S _j (j = 1,..., m) in ℝ^d which are Jordan arcs. We prove that if a system possesses a structure parametrization (ℐ,ϕ) and ℱ(ℐ) is the associated family of ℐ then we have one of the following cases:1. The identity mapping Id does not belong to the closure of ℱ(ℐ). Then (if properly rearranged) is a Jordan zipper.2. The identity mapping Id is a limit point of ℱ(ℐ). Then the arc γ is a straight line segment.3. The identity mapping Id is an isolated point of .We construct an example of a self-similar Jordan curve which implements the third case.Original Russian Text Copyright © 2005 Aseev V. V. and Tetenov A. V.The authors were supported by the Program “Universities of Russia” (Grant UR.04.01.456).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 733–748, July–August, 2005.     

Is the rigid-body assumption reasonable?: Insights into the effects of dynamics on the electrostatic analysis of barnase-barstar

Electrostatically-driven association of proteins is important to many biological functions, and understanding which amino acid residues contribute to these interactions is crucial to protein design. Theoretical calculations that are used to elucidate the role of electrostatics in association are typically based on a single experimentally determined protein structure, while an underlying rigid-body assumption is adopted. The goal of this study was to investigate the role of conformational fluctuations on electrostatic interaction energies, as applied to the electrostatic analysis of barnase-barstar. For our calculations, we apply theoretical alanine-scan mutagenesis to introduce charge perturbations by replacing every ionizable residue with alanine, one at a time. Electrostatic clustering and free energy calculations based on the Poisson-Boltzmann method are used to evaluate the effects of each perturbation. Molecular dynamics simulations are performed for the barnase-barstar parent complex and seven experimental alanine mutations from the literature, in order to introduce relaxation before and after mutation. We discuss the effects of dynamics, in the form of pre- and post-mutation relaxation, on electrostatic clustering and free energies of association in light of experimental data. We also examine the utility of nine electrostatic similarity methods for clustering of barnase alanine-scan mutants. Our calculations suggest that the rigid-body assumption is reasonable for electrostatic calculations of barnase-barstar.     

Approximate symmetry reduction for perturbed nonlinear SchrSdinger equation

We investigate the one-dimensional nonlinear SchrSdinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.     

Affine Weyl group symmetries of Frobenius Painlevé equations

In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.     

Similarity inner solutions for the Pulsar equation

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.     

Similarity analytical solutions for the Schrӧdinger equation with the Riesz fractional derivative in quantum mechanics

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore, the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives α .     

Lie symmetries of a system arising in plasma physics

Lie group classification for a diffusion‐type system that has applications in plasma physics is derived. The classification depends on the values of 5 parameters that appear in the system. Similarity reductions are presented. Certain initial value problems are reduced to problems with the governing equations being ordinary differential equations. Examples of potential symmetries are also presented.