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.     

Exact solutions of some general nonlinear wave equations in elasticity

A similarity analysis of a nonlinear wave equation in elasticity is studied; in particular, one with anharmonic corrections. The symmetry transformation give rise to exact solutions via the method of invariants. In some cases, graphical figure of the solutions are presented. Furthermore, we consider some cases wherein the velocities of the longitudinal and transversal plane waves are variable. Finally, a brief discussion on how a symmetry analysis on a perturbation of the elasticity equation can be pursued.     

On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order

IntroductionWeshallbeconcernedherewiththird-ordersystemofnon-lineardifferentialequationsoftheformX F(X,X)X Ba H(X)=P(t,X,X,X),(l)whereX6R",R"denotestherealn-dimensionalEuclldeanspaceRxRx'xR(nfactors),R=(--ac,ac),Fisannxn-matrixfunction,Bisarealnxn-constantsymmetricmatrix,H:R"~R"1andp:RxR"xR"xR"~R".ThedotsindicatCdifferentiationwithrespeCttot.ThefunctionsF,HandParecontinuousforallvaluesoftheirrespeCtivearguments.Moreover,theexistenceandtheuniquenessofthesolutionsof(l)willbeassu…     

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.     

Oscillatory modes in a nonlinear second-order differential equation with delay

Solution properties of the nonlinear second-order delay-differential equation x(t)=–ax(t)+f[x(t–)] are studied wheref is a piecewise constant function which mimics negative feedback. We show that the solutions can be obtained by a simple geometrical construction which, in principle, can be implemented using a ruler and a compass. Analytical results guarantee the existence and stability properties of limit cycle solutions. Computer-aided constructions reveal a remarkable richness of different types of dynamical behaviors including a variety of unconventional bifurcation schemes.     

Solution of nonlinear wave equation of elastic rod

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia are studied. A nonlinear wave equation is derived. The equation is solved by the method of full approximation.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

Existence of solutions of three-point boundary value problems for nonlinear fourth order differential equation

In this paper, the author uses the methods in [1, 2] to study the existence of solutions of three point boundary value problems for nonlinear fourth order differential equation.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497!\[y^{(4)} = f(t,y,y',y',y')\] with the boundary conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe% qaaiaadEgacaGGOaGaamyEaiaacIcacaWGHbGaaiykaiaacYcaceWG% 5bGbauaacaGGOaGaamyyaiaacMcacaGGSaGabmyEayaagaGaaiikai% aadggacaGGPaGaaiilaiqadMhagaGeaiaacIcacaWGHbGaaiykaiaa% cMcacqGH9aqpcaaIWaGaaiilaiaadIgacaGGOaGaamyEaiaacIcaca% WGIbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaamOyaiaacMcacaGG% PaGaeyypa0JaaGimaaqaaiqadMhagaqbaiaacIcacaWGIbGaaiykai% abg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam4Aaiaa% cIcacaWG5bGaaiikaiaadogacaGGPaGaaiilaiqadMhagaqbaiaacI% cacaWGJbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaam4yaiaacMca% caGGSaGabmyEayaasaGaaiikaiaadogacaGGPaGaaiykaiabg2da9i% aaicdaaaGaayzFaaaaaa!7059!\[\left. \begin{gathered} g(y(a),y'(a),y'(a),y'(a)) = 0,h(y(b),y'(b)) = 0 \hfill \\ y'(b) = b_1 ,k(y(c),y'(c),y'(c),y'(c)) = 0 \hfill \\ \end{gathered} \right\}\] For the boundary value problems of nonlinear fourth order differential equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497!\[y^{(4)} = f(t,y,y',y',y')\] many results have been given at the present time. But the existence of solutions of boundary value problem (*). (**) studied in this paper has not been involved by the above researches. Morcover, the corollary of the important theorem in this paper, i. e. existence of solutions of the boundary value problem of equation (*) with the following boundary conditions.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb% WaaSbaaSqaaiaaicdaaeqaaOGaamyEaiaacIcacaWGHbGaaiykaiab% gUcaRiaadggadaWgaaWcbaGaaGymaaqabaGcceWG5bGbauaacaGGOa% GaamyyaiaacMcacqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGa% bmyEayaagaGaaiikaiaadggacaGGPaGaey4kaSIaamyyamaaBaaale% aacaaIZaaabeaakiqadMhagaGeaiaacIcacaWGHbGaaiykaiabg2da% 9iaadMhadaWgaaWcbaGaaGimaaqabaGccaGGSaGaamOyamaaBaaale% aacaaIWaaabeaakiaadMhacaGGOaGaamOyaiaacMcacqGHRaWkcaWG% IbWaaSbaaSqaaiaaikdaaeqaaOGabmyEayaagaGaaiikaiaadkgaca% GGPaGaeyypa0JaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiqadMha% gaqbaiaacIcacaWGIbGaaiykaiabg2da9iaadMhadaWgaaWcbaGaaG% OmaaqabaGccaGGSaGaam4yamaaBaaaleaacaaIWaaabeaakiaadMha% caGGOaGaam4yaiaacMcacqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaae% qaaOGabmyEayaafaGaaiikaiaadogacaGGPaGaey4kaSIaam4yamaa% BaaaleaacaaIYaaabeaakiqadMhagaGbaiaacIcacaWGJbGaaiykai% abgUcaRiqadogagaGeaiaacIcacaWGJbGaaiykaiabg2da9iaadMha% daWgaaWcbaGaaG4maaqabaaaaaa!7DF7!\[\begin{gathered} a_0 y(a) + a_1 y'(a) + a_2 y'(a) + a_3 y'(a) = y_0 ,b_0 y(b) + b_2 y'(b) = y_1 \hfill \\ y'(b) = y_2 ,c_0 y(c) + c_1 y'(c) + c_2 y'(c) + c'(c) = y_3 \hfill \\ \end{gathered} \] has not been dealt with in previous works.     

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.     

Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity

The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. We present the geometrically nonlinear theory taking into account all possible interaction terms between the elastic and microelastic structures. This is achieved by considering the irreducible pieces of the deformation gradient and of the dislocation curvature tensor. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small. By choosing a particular ansatz we are able to reduce the system of equations to a sine–Gordon type equation which is known to have soliton solutions.     

Self-similar solutions to Lin-Reissner-Tsien equation

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.     

Symmetry classification and optimal systems of a non-linear wave equation

In this paper the complete Lie group classification of a non-linear wave equation is obtained. Optimal systems and reduced equations are achieved in the case of a hyperelastic homogeneous bar with variable cross section.     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations

IntroductionInRef.[1 ] ,theauthorsestablishedtheuniqueexistenceofthesmoothsolutionforthefollowingcouplednonlinearequationsut=uxxx+buux+ 2vvx, (1 )vt=2 (uv) x. (2 )Thesewereproposedtodescribetheinteractionprocessofinternallongwaves.InRef.[2 ] ,ItoM .proposedarecursionoperatorbywhichheinferredthatEqs.(1 )and (2 )possesinfinitelymanysymmetriesandconstantsofmotion .InRef.[3 ] ,P .F .HeestablishedtheexistenceofasmoothsolutiontothesystemofcouplednonlinearKdVequation[4 ]ut=a(uxxx+buux) + 2bvvx,(…     

Solution of Ordinary Differential Equations with a Large Lie Symmetry Group

The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Bäcklund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order.     

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.     

Preliminary group classification of quasilinear third-order evolution equations

Group classification of quasilinear third-order evolution equations is given by using the classical infinitesimal Lie method, the technique of equivalence transformations, and the theory of classification of abstract low-dimensional Lie algebras. We show that there are three equations admitting simple Lie algebras of dimension three. All non-equivalent equations admitting simple Lie algebras are nothing but these three. Furthermore, we also show that there exist two, five, twenty-nine and twenty-six non- equivalent third-order nonlinear evolution equations admitting one-, two-, three-, and four-dimensional solvable Lie algebras, respectively.     

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q ⁽³⁾=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q ⁽³⁾=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q ⁽³⁾==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n5) ODEs which admitnontrivial contact symmetry algebras.