THE STURM-LIOUVILLE-TRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS. L. Trautmann, R. Rabenstein

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1 Worksop on Transforms and Filter Banks (WTFB),Brandenburg, Germany, Marc 999 THE STURM-LIOUVILLE-TRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS L. Trautmann, R. Rabenstein Lerstul für Nacrictentecnik I, University of Erlangen-Nuremberg D-958 Erlangen, Cauerstr. 7, Germany ABSTRACT Partial differential equations (PDEs) are te conventional tool for te description of time- and space-dependent pysical penomena. Vector PDEs arise from te pysical analysis of multidimensional (MD) systems in terms of potential and flux quantities. Expressing te resulting coupled PDEs in vector form facilitates te direct formulation of boundary and interface conditions in teir pysical context. Unfortunately, tese vector PDEs are not suitable for direct computer implementation. We sow ere ow te Laplace transformation wit respect to time and te Sturm-Liouville (SL) transformation wit respect to space lead to multidimensional transfer function models as an equivalent description of te coupled PDEs. Transfer function models are te starting point for te derivation of discrete models by standard metods for one-dimensional systems. Tese discrete models solve te given PDEs numerically, and tey are suitable for computer implemetation. Tis paper explains te transformation of vector PDEs into MD transfer function models. Te telegrap equation is used as an example ere. Te presented SL transformation approac is also suitable for a number of oter tecnical applications, like electromagnetics, optics, and eat and mass transfer.. INTRODUCTION Te conventional tools for te description of time- and spacedependent pysical penomena are partial differential equations (PDEs). Teir derivation usually involves two dependent pysical variables: a potential and a flux quantity. Te relationsip between tese quantities can be establised from te first principles of pysics. Tis results in a pair of coupled PDEs. Tey may be written in a compact form by combining te potential and flux quantities into one vector of unknowns and by arranging te differential operators in matrix notation. Tis form is called a vector PDE. A recently introduced metod for te solution of multidimensional (MD) pysical systems are transfer function models [4]. MD transfer functions are input-output models for pysical penomena wit more tan one independent variable (e.g. time and space). Tey allow one to treat te effects of initial conditions, boundary conditions, and excitation functions on te resulting output signal separately. Since te MD transfer function model contains bot, potential and flux, boundary and interface conditions can be formulated directly in teir pysical context. Transfer function models are an excellent starting point for te development of discrete-variable representations of pysical systems. Tey offer te following advantages: suitability for computer implementation by non-iterative algoritms, stability in te sense of systems teory, and ig numerical accuracy witout proibitively small step sizes [3]. Te crucial point in te derivation of MD transfer function models is te coice of suitable transformations for te independent variables. For pysical systems described by scalar PDEs, Laplace transformation wit respect to time and Sturm-Liouville(SL) transformation wit respect to space ave been sown to be appropriate [3, 4]. Tis paper introduces an SL transformation for te MD transfer function description of vector PDEs. Te conversion of a vector PDE into a MD transfer function model is described step-by-step wit te example of te telegrap equation. Te paper is organized as follows: Section describes te general form of a vector PDE wit initial and boundary conditions. An example is given by te telegrap equation. Section 3 gives an outline for te solution of tis vector PDE. Section 4 discusses te Laplace transformation wit respect to te time variable. Section 5 sows te construction of te SL transformation for te space variable. Te resulting transfer function model is presented in Section 6, and its discretization is discussed in section 7. Section 8 presents a sort example.

2 . PROBLEM DESCRIPTION We consider a system of coupled PDEs wit time t and one space dimension x of te generic vector form Ay(x; t) + BD x y(x; t) + CD t y(x; t) : () D x denotes derivation wit respect to te space variable and D t derivation wit respect to time. Te matrices A, B and C contain loss terms, terms for spatial derivatives and capacitance elements, respectively. Te vector y(x; t) includes bot te potential u(x; t) and te flux quantity i(x; t) y(x; t) u(x; t) i(x; t) : () Te initial conditions for tis vector PDE are given by y(x; ) y i (x) : (3) At te endpoints of te interval x < x < x, we set boundary conditions by a suitable combination of potential and flux, expressed by te boundary operator f b (H denotes matrix conjugate transpose). f H b y(x n; t) n (t); n ; : (4) Example: Te well-known telegrap equation as voltage u(x; t) and current i(x; t) as potential and flux quantities. It is given by (,) wit te line parameters c, l, r, g and A g r ; B ; C c l : (5) We consider omogeneous initial conditions (6) and firstorder boundary conditions wit given voltage (7) y i (x) ; (6) f b : (7) 3. OUTLINE FOR THE SOLUTION OF THE VECTOR PDE Te given PDE () wit initial conditions (3) and boundary conditions (4) can be solved te following way:. Laplace transformation wit respect to time to remove te time derivatives and to include te initial conditions as an additive term.. Finding a SL transformation for te space variable tat removes te spatial derivatives and includes te boundary values as an additive term. 3. Writing te resulting algebraic equation as a transfer function model. 4. Discretizing te transfer function model for computer implementation. Tese steps are carried out in te next sections. 4. TRANSFORMATION WITH RESPECT TO TIME In te first step, we apply te Laplace transformation (8) wit te temporal frequency variable s and make use of te differentiation teorem (9) Lfy(x; t)g Y(x; s) Z e?st y(x; t) dt ; (8) LfD t y(x; t)g sy(x; s)? y(x; ) : (9) Tis removes te time derivatives in () and turns te initialboundary value problem (,3,4) into te boundary value problem (,) by introducing te initial value (3) as an additive term into (). AY(x; s) + BD x Y(x; s) + scy(x; s) Cy i (x) () f H b Y(x n; s) n (s); n ; () 5. STURM-LIOUVILLE TRANSFORMATION In te second step, we construct a functional transformation wic as similar properties for te space variable as te Laplace transformation as for te time variable: It sall remove te space derivatives described by te differential matrix operator BD x and it sall include te boundary values (4) into te resulting algebraic equation. It is known from scalar PDEs tat tere is no unique transformation tat suits all possible cases. Instead, te transformation kernel as to be adapted to te matrix A, te matrix operator BD x and te boundary value operator f b of te specific PDE at and. We sow ere ow to perform tis process. Te dependence on te temporal frequency variable s is omitted, since it is independent of te spatial transformation. 5.. Transformation formulation We want to find a transformation formulation for te space variable wit te same structure tat (8) as for te time variable. According to te different integration limits (x and x instead of zero and infinity), we ave to find a specific transformation kernel vector (x; ~ ) for te transformation of te desired vector PDE and its boundary conditions. Bot te transformation kernel and te spatial frequency variable ~ are yet to be determined depending on te matrices A, B, C and te boundary conditions. At first, we develop te formulation of te transformation by considering te term CY(x) in (). Te influence of te matrices A and B is discussed in te next section. Multiplying CY(x) from te left and side wit te conjugated transponsed vector H (x; ~ ) and integrating over x constitutes a scalar product for H (x; ~ ) and CY(x).

3 It defines te transformation wit respect to te space variable x T fcy(x)g Y ( ~ ) 5.. Differentiation teorem H (x; ~ )CY(x)dx : () We now want to derive a differentiation teorem for te spatial transformation wic as properties similar to (9) for te Laplace transformation: te spatial derivatives sould be removed and te boundary conditions sould be inserted as an additive term after spatial transformation. First we combine te loss matrix A and te operator BD x into a single operator L L A + BD x : (3) Inserting (3) into () we ave to transform LY(x; s) into te spatial frequency domain. L includes spatial derivatives so tis is similar to te left and side of (9) were time derivatives exist. To obtain a spatial differentiation teorem wit te same structure as te temporal differentiation teorem (9), we use te approac (4). It contains no spatial derivatives and includes te boundary values b ( ~ ) in te spatial frequency domain T fly(x)g ~ Y ( ~ )? b ( ~ ) : (4) Tis is acieved by te following steps. Wit partial integration we obtain T fly(x)g H (x; ~ )LY(x)dx? A H? B H D x ~K(x; ~ )dx + Y H (x) + Y H (x)b H K(x; ~ ~ ) Wit te boundary values written as i Y H B K ~ x and wit te operator i x : (5) b g H b Y? ~gh b f H b Y i x (6) ~L A H? B H D x ; (7) we obtain (8) for te first and second summand of () after spatial transformation. H LYdx Y H ~ L dx+ + b g H b Y? ~gh b f H b Y i x (8) Tis is an extended Green's formula were L ~ is te adjoint operator to L []. Te separation of Y H B K ~ into fb, ~ fb, g b, and ~gb in (6) is not unique. It can be adapted to te kind of boundary conditions, suc tat f H b Y corresponds to (7) Determination of te transformation Kernel Following te desired differentiation teorem of te spatial transformation (4) we ave to find a transformation kernel ~K(x; ~ ) tat complies wit () and (). Ten Green's formula (8) can be written in terms of (4) wit b ( ~ ; s) ~g H b (x; ~ ) i x : (9) ~L (x; ~ ) ~ C H (x; ~ ) () ~ f H b (x n ; ~ ) ; n ; () Tis is a generalization of te Sturm-Liouville type problem and tat is te reason for calling te spatial transformation a Sturm-Liouville transformation. Te elements of te transformation kernel are te eigenfunctions of te operator ~L. Te eigenvalues are described by te discrete frequency variable ~. After spatial transformation we can write () as one algebraic equation (), wic includes te boundary values of () as an additive term. s Y( ~ ; s) + ~ Y( ~ ; s) b ( ~ ; s) + y i ( ~ ) () Example: For te telegrap equation te transformation kernel vector results wit () and () in ~K(x; ~ ) 4 (r? ~ l)a sin(xa) cos(xa) 3 5 (x) : (3) Te discrete spatial frequency variable can be calculated from (r? ~ l)(g? ~ c) + : (4) A 5.4. Inverse transformation Since te elements of (x; ~ ) are not necessarily ortogonal, we ave to find a transformation kernel K(x; ) for te inverse transformation tat is biortogonal to (x; ~ ). K results from te adjoint eigenvalue problem of () and (): LK(x; ) CK(x; ) (5) f H b K(x n; ) ; n ; (6) K and K ~ are biortogonal functions wit respect to te weigting matrix C. Tis can be sown by applying Green's formula. Replacing Y in (8) wit K we obtain H ( ~ )LK( )dx? b g H b K? ~gh b f H b K i x (7) K H ( ) ~ L ( ~ )dx

4 Te rigt-and side of (7) must be zero wit respect to () and (6) and te left-and side of (7) can be evaluated to (8) wit respect to () and (5). H ( ~ )LK( )dx?? ~ K H ( ) ~ L ( ~ )dx H ( ~ )CK( )dx (8) Since te first factor of te rigt and side of (8) is only zero for ~, te second factor must be zero for ~ 6. Wit te relationsip between te operators L and L ~ it can be sown tat teir eigenvalues are related by ~. From (4) we can see tat every eigenvalue ~ as one conjugate counterpart, indexed ere wit. It follows tat ~ ~. Tus we can formulate K(x; ~ ) in terms of te eigenvalues ~ of K(x; ~ ~ ). In tis notation te integral of te rigt and side of (8) must be zero for 6. Ten K and K ~ are biortogonal functions wit respect to te weigting matrix C. For inverse transformation we only ave to sum te scalar frequency values Y ( ~ ; s) weigted wit K(x; ~ ) and a frequency dependent norm factor N since te eigenvalues are discrete Y(x; s) T? f Y ( ~ ; s)g X? N Y ( ~ ; s)k(x; ~ ) : (9) N can be evaluated by inserting (9) into (): Y ( ~ ) Z x Z x K ~ H ( ~ )CYdx X K ~ H ( ~ )C? X Z x Y ( ~ ) N? N Y ( ~ )K(x; ~ )dx K ~ H ( ~ )CK(x; ~ )dx (3) Since and K form a biortogonal system, (3) as only non-zero elements for. Tus N is N Z x K ~ H ( ~ )CK(x; ~ )dx : (3) Example: In te telegrap equation wit first order boundary conditions wit given voltage te eigenvectors are K(x; ~ ) 4?(r? ~ sin(xa) l)a cos(xa) 3 5 K (x) (3) and te eigenvalues can be evaluated from (r? ~ l)(g? ~ c) + : (33) A N follows from (3) as N ~ lc? rc? gl ~ c? g A : (34) 6. TRANSFER FUNCTION MODEL AND INTERPRETATION AS FILTER Once te algebraic equation (3) is obtained, te tird step is quite straigtforward. Solving (3) for Y ( ~ ; s) te PDE wit boundary and initial conditions can be written as a set of transfer functions in te temporal and spatial frequency domain. Y ( ~ ; s) G b ( ~ ; s) b ( ~ ; s) + + G i ( ~ ; s)y i ( ~ ) (35) Te inputs are te terms y i ( ~ ) and b ( ~ ; s), te output is te transformed variable Y ( ~ ; s) of te desired solution y(x; t) of te PDE. Te inputs are multiplied wit te transfer functions G b ( ~ ; s) G i ( ~ ; s) s + ~ (36) Te transfer function model (35) can also be interpreted as a MD filter wit te temporal frequency variable s and te spatial frequency variable ~. Tus any kind of excitation results only in te eigenfrequencies of te oscillating system. Tis represents te pysical background of te PDE. 7. DISCRETIZATION In tis section we turn te transfer function description of te PDE (35) into a discrete-time and discrete-space model. Since te procedure is analogous to te case of scalar PDEs, a roug sketc is sufficient [5]. 7.. Discretization wit respect to time Wit respect to te temporal frequency variable s, te transfer functions in (36) correspond to first-order one-dimensional systems. A discrete-time model can tus be obtained by standard analog-to-discrete transformations. Teir application to scalar PDEs as been described in [3] and applies in te same fasion to vector PDEs.

5 7.. Discretization wit respect to space Space discretization is performed by application of te inversion formula (9) to te output of te discrete-time models described above (see [3, Fig. ]). Te summation in (9) is truncated to a reasonable number of terms and evaluated at discrete points x n in te space of interest. In contrast to oter metods for te numerical solution of PDEs, te accuracy does not depend on te number and spacing of te spatial gridpoints. Tis means tat te numerical load can be restricted to only tose points tat are actually of interest. If te solution is required at only one point in space, ten it may be computed at tis point only wit te same accuracy as for a ig number of gridpoints. 8. EXAMPLE In tis example we examine te voltage distribution along an electrical transmission line. Te lengt of te transission line sould be x? x m, te serial resistance is r /m, te sunt conductance is g 4 ms/m, te sunt capacitance is c 4 F/m and te serial inductance is l mh/m. Te transmission line is activated by an impulse of lengt : ms at te boundary x. Te result displayed in Fig. as been obtained wit a discrete model of (35) and wit an inverse transformation (9) truncated to 65 terms. Te effects of wave propagation, damping and reflection at te boundaries are correctly reproduced in te simulation. Te results compare favourably wit tose obtain by numerical integration according to te MD wave digital principle []. u(x,t) [V] t [ms].5 x [m] Figure : Voltage distribution along a transmission line general framework for te description of pysics-based multidimensional systems. Altoug presented ere for te example of te telegrap equation it can be used for a wide variety of PDEs. It is also applicable to two or tree spatial dimensions wit a more general form of Green' s formula. Higer-order differential equations and space-dependent coefficients are considered wit appropriate definitions of te matrix operator L and te weigting matrix C. Besides boundary and initial values, external excitation functions can also be included. Tis generality makes te SL transformation approac suitable for a number of tecnical applications, like electromagnetics, optics, and eat and mass transfer.. REFERENCES [] R.V. Curcill. Operational Matematics. McGraw- Hill, New York, 3. edition, 97. [] H. Krauss, R. Rabenstein, M. Gerken. Simulation of wave propagation by multidimensional digital filters. Simulation Practice and Teory, 4:36-38, 996 [3] R. Rabenstein. Discrete simulation models for multidimensional systems based on functional transformations. In J.G. McWirter and I.K. Proudler, editors, Matematics in Signal Processing IV. Oxford University Press, 998. [4] R. Rabenstein. Transfer function models for multidimensional systems wit bounded spatial domain. Matematical and Computer Modelling of Dynamical Systems, 5(), 999. In print. [5] R. Rabenstein, L. Trautmann. Solution of Vector Partial Differential Equations by Transfer Function Models. IEEE Int. Symp. on Circuits and Systems (ISCAS). Orlando, USA, CONCLUSIONS Te presented Sturm-Liouville (SL) transformation metod for te conversion of vector partial differential equations (PDEs) into transfer function models constitutes a rater