Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

弹性理论方程的不变解

基于李群和李代数理论,分析了固体力学中微分方程的群分析的基本原理和应用.总结了群分析在弹性理论领域取得的一些重要成果,特别是弹性动力学中的拉梅方程和非线性弹性理论方程方面,得到了弹性理论方程的一系列不变解.预测了群分析在弹性理论领域的进一步发展方向.     

Approximate Group Analysis and Multiple Time Scales Method for the Approximate Boussinesq Equation

This paper is devoted to investigation of the approximate Boussinesq equation by methods of the approximate symmetry analysis of partial differential equations with a small parameter developed by Baikov, Gazizov and Ibragimov. We combine these methods with the method of multiple time scales to extend the domain of definition of approximate group invariant solutions of the approximate Boussinesq equation.     

Continuation of Approximate Transformation Groups via Multiple Time Scales Method

Recently, the theory of approximate symmetries was developedfor tackling differential equations with a small parameter. This theoryfurnishes us with a tool, e.g. for constructing approximate groupinvariant solutions. Usually, these solutions are determined by powerseries in the small parameter and hence they are well defined only in asmall region of independent variables. In this paper, we modify theapproximate symmetry analysis by combining it with the multiple timescales method. In this way, we can extend the domain of definition ofapproximate symmetries of differential equations with a small parameterand of their invariant solutions. The method is illustrated by the vander Pol equation. It is shown that, in this example, our approachprovides a group theoretical background of ad hoc methods widelyused in perturbation techniques.     

An Existence Theorem for Generalized von Kármán Equations

Using techniques from formal asymptotic analysis, the first two authors have recently identified generalized von Kármán equations, which constitute a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, the remaining portion being free. In this paper, we establish an existence theorem for these equations. To this end, we first reduce them to a single equation, which generalizes a cubic operator equation introduced by M.S. Berger and P. Fife. We then directly solve this equation, notably by adapting a crucial compactness method due to J.-L. Lions. Résumé. En utilisant les techniques de l'analyse asymptotique formelle, les deux premiers auteurs ont récemment identifié des équations de von Kármán généralisées, qui constituent un modèle bi-dimensionnel de plaque non linéairement élastique dont une partie seulement de la face latérale est soumise à des conditions aux limites de von Kármán, la partie restante étant libre. Dans cet article, on établit un théorème d'existence pour ces équations. À cette fin, elles sont d'abord réduites à une seule équation, qui généralise une équation faisant intervenir un opérateur cubique, introduite par M.S. Berger et P. Fife. On résout ensuite directement cette équation, en adaptant notamment une méthode cruciale de compacité due à J.-L. Lions.     

Reproduction of solutions of bidimensional ideal plasticity

The symmetries of a system of differential equations allowed the transformation of its solutions to a solution of this system. New analytical exact solutions of a system of two-dimensional ideal plasticity equations were constructed from two well-known solutions, that for a circular cavity stressed by normal pressure, and Prandtl's solution for a block compressed between perfectly rough plates, for the case where the thickness of the block was rather small. A mechanical sense of new solutions was discussed.     

On Non-Symmetric Deformations of Neo-Hookean Solids

Deformations possible (i.e., those satisfying the governing three-dimensional equations of equilibrium and the incompressibility constraint) within a class of non-symmetric deformations for a neo-Hookean nonlinearly elastic body were determined in [1], where it was found that only three special cases of the class of deformation fields considered could be solutions. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation contained within a family of universal deformations. In this paper, the results reported in [1] are shown to hold for a substantially broadened deformation field. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fundamental Solutions for Zero-Coupon Bond Pricing Models

The transformation approach is employed to reduce one-factor bond-pricing equations into the heat equation for which the fundamental solution is wellknown. These transformations are subsequently used to construct the fundamental solutions of two zero-coupon bond-pricing equations. The closed-form analytical solutions of the Cauchy initial value problems of the two bond-pricing model equations are then obtained.     

Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation

Exact solutions for a class of nonlinear partial differential equations modelling soil water infiltration and redistribution in irrigation systems are studied. These solutions are invariant under two-parameter symmetry groups obtained by the group classification of the governing equation. A general procedure for constructing invariant solutions is presented in a way convenient for investigating numerous new exact solutions.     

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

All irreducible regular partially invariant submodels with one noninvariant function for the equations of ideal magnetohydrodynamics are constructed. The submodels are completed to involution, and partially integrated. The submodels specify Ovsyannikov vortex type motion or motion with homogeneous deformation in some spatial directions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 5–15, March–April, 2009.     

On the Solution of the Unforced Damped Duffing Oscillator with No Linear Stiffness Term

Using a series of functional transformations we reduce the unforced,damped Duffing oscillator to equivalent equations of the Abel andEmden–Fowler classes. Taking into account the known exact analyticsolutions of these equivalent equations we prove that there does notexist an exact analytic solution of the damped, unforced Duffingoscillator in terms of known (tabulated) analytic functions. It followsthat a new class of solutions must be defined for solving this problem`exactly'. Finally, a new approximate solution of the intermediateintegral of the damped Duffing oscillator with weak damping isconstructed.     

An Integrable Model of Nonstationary Rotationally Symmetrical Motion of Ideal Incompressible Liquid

We consider a partially invariant solution of the Eulerequations with respect to a six-parameter Lie group admitted by thissystem where the vertical component of velocity is a function of thevertical coordinate and time only while two other components andpressure do not depend on the polar angle in a cylindrical coordinatesystem. The analysis of the corresponding overdetermined system leads totheir special (but nontrivial) dependence of the polar radius. Afterthis, the nonlinear factor-system for invariants of the group is reducedto a system of ordinary differential equations by introduction ofLagrangian coordinates. As a result, we obtain a wide class of new exactsolutions which describes vortex motions of an ideal incompressibleliquid including motions with singularities.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

On Bifurcation in Finite Elasticity: Buckling of a Rectangular Rod

Although there is an extensive literature on the linearization instability of the nonlinear system of partial differential equations that governs an elastic material, there are very few results that prove that a second branch of solutions actually bifurcates from a known solution branch when the known branch becomes unstable. In this paper the implicit function theorem in a Banach space setting is used to prove that the quasistatic compression of a rectangular elastic rod between rigid frictionless plates leads to the buckling of the rod as is observed in experiment and as first predicted by Euler. This work was supported in part by the National Science Foundation under Grant No. DMS–8810653 and DMS–0405646.     

Anti-plane Shear Deformations in Compressible Transversely Isotropic Materials

Anti-plane shear deformations in a compressible, transversely isotropic hyperelastic material are under investigation. The displacement is assumed to be along the direction of the symmetry axis and is independent of the axial position. The resulting equations of equilibrium form an overdetermined system of partial differential equations for which solutions do not exist in general. Necessary and sufficient conditions are derived for such materials to sustain anti-plane shear deformations in the sense that every solution to the axial equation automatically satisfies the other two in-plane equations. Comparison is made with results for isotropic materials. A weaker version of the conditions specialized to axisymmetric anti-plane shear deformations is also obtained.     

Symmetry Solutions for Transient Solute Transport in Unsaturated Soils with Realistic Water Profile

We examine solutions for solute transport using the convection-dispersion equation (CDE) during steady evaporation from a water table. It is common, when solving the CDE, to first approximate the volumetric water content of the soil as a constant. Here, we assume a reasonable function for the water content profile and construct realistic nonlinear hydraulic transport properties. Both classical and nonclassical symmetry techniques are employed. Invariant solutions are obtained for the one dimensional CDE even with a nontrivial background profile for volumetric water content.