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Fokas's Unified Transform Method for Linear Systems

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Fokas's Unified Transform Method for Linear Systems QUARTERLY OF APPLIED MATHEMATICS VOLUME LXXVI, NUMBER 3 SEPTEMBER 2018, PAGES 463–488 http://dx.doi.org/10.1090/qam/1484 Article electronically published on September 28, 2017 FOKAS’S UNIFIED TRANSFORM METHOD FOR LINEAR SYSTEMS By BERNARD DECONINCK (Department of Applied Mathematics, University of Washington, Campus Box 353925, Seattle, WA, 98195 ), QI GUO (Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095-1555 ), ELI SHLIZERMAN (Department of Applied Mathematics & Department of Electrical Engineering, University of Washington, Campus Box 353925, Seattle, WA, 98195 ), and VISHAL VASAN (International Centre for Theoretical Sciences, Bengaluru, India) Abstract. We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial dif- ferential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, as they arise symmetrically in the global relations. This allows the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. Wave equations are treated separately in an appendix. 1. Introduction. The Unified Transform Method (UTM) or Method of Fokas pre- sents a new approach to the solution of boundary value problems (BVPs) for integrable nonlinear equations [6,8], or even more successfully for linear, constant coefficients partial differential equations (PDEs) [5, 8]. In this latter context, the method has resulted in new results for one-dimensional problems involving more than two spatial derivatives [7,8], for elliptic problems [3,8], and for interface problems [4,11], to name but a few. A more complete picture can be obtained from the online repository at [1]. Fewer studies exist of how the method applies to linear systems of equations or to higher-order scalar evolution equations that may be rewritten as such. In fact, to our knowledge, the application of the UTM to the standard wave equation presented in the appendix here is new, albeit not surprising. Systems of equations are examined in Received April 28, 2017, and, in revised form, August 10, 2017. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected] c 2017 Brown University 463 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 464 BERNARD DECONINCK, QI GUO, ELI SHLIZERMAN, AND VISHAL VASAN [9,13–15] and [16]. In [16], a problem from thermoelastic deformation is examined, while in [9] two water wave-related systems are treated. In [15], the UTM is applied to the wave equation in a moving domain, while in [13, 14] the free Klein-Gordon equation is examined. The treatment in these papers is non-generic in the sense that the different branches of the dispersion relations of the systems considered are either polynomial or rational in the wave number (as in [9, 15]) or are conveniently parameterized to avoid radicals, as in [13, 14, 16]. In addition, all five papers start from a Lax pair and con- struct a Riemann-Hilbert problem almost exclusively, with almost no hints as to how to generalize the more accessible UTM for scalar systems based on the use of Green’s Theorem. Generically, the different branches of the dispersion relations for a first-order evolution system of dimension N depend on radicals as they are the roots of an N-th order polynomial. Below we investigate the general case of an N-dimensional first-order linear system, using two examples. The first example is the (free: no potential) Klein-Gordon (KG) equation [10, 13, 15] utt − uxx + αu =0, (1) where α is a constant parameter and indices denote partial derivatives. For our purposes, the KG equation is rewritten as a two-dimensional first-order system: qt = p, (2a) pt = qxx − αq, (2b) where q = u, p = ut. Our approach to the KG equation differs from [13, 15] in that all our considerations are based on so-called local relations, to which Green’s Theorem is applied. No parameterization of the dispersion relation branches is used. Rather we use the branches of the dispersion relation in their original form. Lastly, in [13, 14] only α>0 is considered. The second example is the linearized Fitzhugh-Nagumo (FN) system of partial differ- ential equations [12] vt = vxx − v − w, (3a) wt = βv, (3b) where β is a constant parameter. In general, we consider systems of the form Qt +Λ(−i∂x)Q =0, (3) where Q is an N-dimensional vector and Λ is a matrix-valued polynomial of size N × N and of order n. Example 1. For the KG equation, we have q 0 −1 q ∂t + − 2 =0, (4) p ∂x + α 0 p so that n =2,and − − 0 1 Λ( i∂x)= 2 . (5) (−i∂x) + α 0 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. FOKAS’S UNIFIED TRANSFORM METHOD FOR LINEAR SYSTEMS 465 Example 2. Similarly, for the FN system, v −∂2 +1 1 v ∂ + x =0, (6) t w −β 0 w allowing us to read off Λ(−i∂x)andn = 2 as well. Throughout, we contrast the general systems case with the general scalar case, where theequationcanbewrittenintheform ut + λ(−i∂x)u =0, (7) for a scalar-valued function u(x, t). Although different non-local systems can be considered by allowing Λ to depend ra- tionally on its argument (see e.g. [17] for the scalar setting), in this paper we restrict to the case where Λ (and λ) depends polynomially on its argument. As a consequence, the dispersion relation for the scalar case, easily found by equating u =exp(ikx − ωt), is given by ω = λ(k), and ω depends polynomially on k. It should be remarked that in the above calculation of the dispersion relation, we have followed the convention in the literature on the UTM for the dispersion relation. Thus the dispersion relation as used here differs by a factor of i from the standard use. For instance, in the UTM dispersive equations are characterized by a purely imaginary dispersion relation for k ∈ R. Similarly, for the systems case (3), we let ⎛ ⎞ Q1 ⎜ . ⎟ ikx−ωt Q = ⎝ . ⎠ e , (8) QN so that ω satisfies det(Λ(k) − ωI)=0, (9) where I is the N × N identity matrix. Thus the different branches Ω1, ..., ΩN of the dispersion relation are roots of an N-th order polynomial. Generically, they depend on radicals of order up to N (N-th roots) whose arguments are polynomials in k. It follows that these dispersion branches are sheets of an N-valued function and they have branch point singularities in the complex k plane. This is not always the case, as is illustrated in the appendix for the wave equation and in the examples treated in [9]. Throughout this paper, we assume that all branches Ωj (k) are distinct, except at isolated values of k ∈ C. Example 1. For the KG equation, 2 Ω1,2 = ±i α + k , (10) which has branch points at the square roots of −α. We choose the branch cut that connects these branch points. As is detailed in what follows, this choice is inessential and does not affect the results obtained. Example 2. For the FN system, 1+k2 ± (1 + k2)2 − 4β Ω = , (11) 1,2 2 which has 4 branch points. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 466 BERNARD DECONINCK, QI GUO, ELI SHLIZERMAN, AND VISHAL VASAN One of the main advantages of the UTM is its ability to characterize the number and the type of boundary conditions required to ensure wellposedness of a given initial- boundary value problem [8]. We wish to see to what extent the same can be done for systems of PDEs. To this end, we do not specify boundary conditions for our examples at this point. Rather, this is done at a later stage in this paper. We will see how the application of Fokas’s UTM determines the information that should be provided on the boundary of our domain. We limit ourselves to problems posed on the half line x>0. It is anticipated that extending our results to problems on the finite interval x ∈ (0,L)is comparable to extending the UTM for scalar problems on the half line to scalar problems on the finite interval. Of course, in all cases, we specify initial conditions Q(x, 0) = Q0(x). In the next sections, we go through the extension of Fokas’s UTM, as applied to systems of linear, constant-coefficient evolution PDEs. Each section deals with a different step of the method, so as to present the method in an algorithmic way. 2. The local relation. The first step for the application of the UTM is to rewrite the system of equations in divergence form. We refer to this form as the local relation. Following the appendix of [9], we rewrite (3) as e−ikxI+Λ(k)tQ − e−ikxI+Λ(k)tX(x, t, k)Q = 0 (12) t x ⇒ e−ikxI+Ω(k)tA(k)Q − e−ikxI+Ω(k)tA(k)X(x, t, k)Q =0, (13) t x where Ω(k) = diag(Ω1,...,ΩN ), the diagonal matrix with the different branches of the dispersion relation as diagonal elements. The matrix A(k) diagonalizes the matrix Λ(k): Λ(k)=A−1(k)Ω(k)A(k).
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