On the discretization timestep in the finite element. Sara perna submitted on 12 oct 2015 v1, last revised 7 sep 2016 this version, v2. As examples, kneser found the bilinear expansion for rectangular and circular areas and for the surface of a sphere. The different approaches used in developing one or two dimensional heat equations as well as the applications of heat equations. Unfortunately, this is not true if one employs the ftcs scheme 2. As we said then, the extent to which we can express modular forms in terms of. When the argument is shifted by the vectors 2me k, the riemann theta function f k is transformed according to the law 1. Together with the heat conduction equation, they are sometimes referred to as the evolution equations because their solutions evolve, or change, with passing time. We are interested in properties of coefficients of certain expansions of the classical theta functions. As an application, we prove that two remarkable spaces of vectorvalued modular forms, which seem to be unrelated at a first look since they are constructed in. As an application, we prove that two remarkable spaces of vectorvalued modular forms, which seem to be unrelated at a first look since they are constructed in two very different ways, are the same. However, i n contrast to th e abe l case they are n o t in d e p e n d e n t. Applications of the jacobi theta functions include the analytic solution of the heat equation, square potential well problems in quantum mechanics, wannier functions in solid state physics, conformal mapping of periodic regions, gravitational physics, quantum cosmology, coding theory, sphere packings, crystal lattice calculations, and study of. My objective here will be to provide a concise account of the stark essentials of some of my recent work as it relates to that wonderful creation of the youthful jacobithe theory of theta functions.
More precisely, the quotient of hypergeometric theta functions by a. In those sections, we made careful study of theta functions with real parameter t, which we now see recovered by setting z it. Theta method is used to deal with fractional telegraph differential equation defined by caputo fractional derivative for different values of. One can show that the exact solution to the heat equation 1 for this initial data satis es, jux.
Differential equations involving cubic theta functions and eisenstein. The main feature of an euler equation is that each term contains a power of r that coincides with the order of the derivative of r. It is getting a bit far from the material of this course, but i wanted to write it up here so that you can see the connection to. One jacobi theta function named after carl gustav jacob jacobi is a function defined for two complex variables z and.
There are several closely related functions called jacobi theta functions, and many different and incompatible systems of notation for them. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting. Also assume that heat energy is neither created nor destroyed for example by chemical reactions in the interior of the rod. Plugging a function u xt into the heat equation, we arrive at the equation xt0. Heat equations and their applications one and two dimension. Heat equation for theta functions and vectorvalued. The resulting operator is identified with a blend of the inozemtsev hamiltonian and the. Elliptictheta2 for fixed z the mathematical functions site. David mumfords tata lectures on theta volumes are an excellent source of reading on this topic. Heat equation for theta functions and vectorvalued modular forms authors. We therefore obtain a product formula for the theta function in the form. The reader who is unfamiliar with the details of the following argument should return to this section in due course and, meanwhile, accept equations 1. Note on trigonometric expansions of theta functions.
Applied theta functions of one or several variables nicholas wheeler, reed college physics department october 1997 introduction. All four theta functions are entire functions of v. The heisenberg group and theta functions with 5 characteristics 4. Thetanulls and differential equations semantic scholar. If fz is an analytic function on d, c is a simple counterclockwise contour in d, then z c. We can choose any of these four theta functions and then define the remaining three in terms of the one chosen. Heat equation u tu xx finite difference scheme theta method. But this writeup will brie y describe how the theta function solves the heat equation. Pdf numerical solution of fractional telegraph differential. Jacobi subjected theta functions to a systematic investigation, and picked out four special theta functions, which formed the basis of his theory of elliptic functions cf. The main feature of an euler equation is that each term contains a power of r.
So, it is reasonable to expect the numerical solution to behave similarly. In a way, the main theme underlying the whole paper is the heat equation, which allows us to. Notes on the poisson summation formula, theta functions, and. Contemporary mathematics heat kernels, theta identities, and zeta. Dividing this equation by kxt, we have t0 kt x00 x. This shows that the heat equation respects or re ects the second law of thermodynamics you cant unstir the cream from your co ee. Theta functions are also useful in physics for solving the heat equation. Tata lectures on theta i division of applied mathematics brown. Individual examples of theta functions are already encountered in the work of j. Solution of the heatequation by separation of variables. In summary then, by 1950 there were essentially three methods16 for. This gives rise to the action of the modular group on theta functions.
The jacobi theta function is the fundamental solution of the onedimensional heat equation with. Fuchs equations of elliptic modular surfaces, and following jacobis idea, derived a. Bellman points out that theta reciprocity is a remarkable consequence of the fact that when the theta function is extended to two variables, both sides of the. Well use this observation later to solve the heat equation in a. We show that they are solutions of a differential system derived from the heat equation. Doubly periodic meromorphic functions via kz,t 24 7. The jacobi theta functions are essentially rotations in a complex elliptic space and as such provide a base for the lattices of the exceptional lie algebras e6 and e 8 in complex 3space and. On the discretization timestep in the finite element thetamethod 567 the problems with mixed boundary conditions newton and dirichlet, see e. Numerical methods for solving the heat equation, the wave. Abelian theta functions c curve riemann surface of genus g. Theta functions in general are powerfully versatile. Theta functions and nonlinear equations 15 the vectors f k can also be written in the form 1. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
David mumfords tata lectures on theta volumes are an excellent source of reading on this. For the circle rz we can calculate the theta function explicitly. The work of bringmann and lovejoy 8, 16 studies the series f. It is getting a bit far from the material of this course, but i wanted to write it up here so that you can see the connection to the heat and schr odinger equations on the circle. When the argument is shifted by the vectors 2me k, the riemann theta function f k is transformed according to the. Note on trigonometric expansions of theta functions sciencedirect. Heat equation for theta functions and vectorvalued modular. There are two particularly elegant examples of mock theta functions with shadow proportional to weight 12 unary theta functions. Notes on the poisson summation formula, theta functions.
We can start only from halphens equation and the heat equation in order to study theta functions form analytic vieqpoint. We shall introduce the theta functions by considering a specific heat conduction problem. Fourier analysis in polar and spherical coordinates. We give a method for constructing vectorvalued modular forms from singular scalarvalued ones of a suitable type.
The theta function satisfies the following heat equation. Another motivation for looking at theta and eta functions comes from complex multiplication of elliptic curves. What do theta functions have to do with quadratic reciprocity. In mathematics, theta functions are special functions of several complex variables. We plan to explicitly give expressions of these coefficients.
Other functions that can be expressed in terms of theta functions include eisenstein series g 2k. Observe that this function extends meromorphically to all of c with rst order poles in the set iz. On the discretization timestep in the finite element theta. Second order linear partial differential equations part iv. Heat equation u tu xx finite difference scheme theta. A di erentialdi erence operator is used to model the heat equation on a nite graph analogue of poincar es upper halfplane. In 85 of chapter i we introduced two of these functions for which we used the notation dx, t and q x, t. The theta function proof is also discussed in dym and mckeans 1972 book fourier series and integrals and in richard bellmans 1961 book a brief introduction to theta functions. We observed that they satisfy the heat equation and that they are periodic in x. In fact, wed like to use the symmetry property of the realvalued theta function as t7.
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