Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods

Mathl. Comput. ModellingVol. 26, No. 4, pp. 13-34, 1997

Pergamon

PII: SOS957177(97)00142-S

Nonlinear Models in Applied Sciences Quadrature to Collocation

Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/97 $17.00 + 0.00

and Problems from Differential Generalized Met hods

N. BELLOMO Department of Mathematics, Politecnico di Torino 10129 Torino, Italy Dedicated to Professor F. Ziegler on his 60th birthday. (Received June 1997; accepted July 1997)

Abstract-This paper deals with the developments of mathematical methods for the discretization of continuous models and the solution of nonlinear problems of interest in applied sciences. In particular, the contents refer to developments of the differential quadrature method proposed by Bellmann, Kashef and Cssti, which leads to the so called generalized collocation methods. The contents is in three parts. The first one is a general description of the method for the solution of initial-boundary value problems. The second part is on recent developments of the method both towards the solution of different classes of problems, e.g., solution of integro-differential equations, domain decomposition and stochastic problems. The third part is on improvements of solution algorithms, on computation of error estimates, and research perspectives. The whole content is constantly referred to the solution of nonlinear problems in applied sciences. Keywords-Nonlinear cation, Interpolation,

problems, Nonlinear sciences, Evolution equations, Sine functions, Spectral methods.

Collo-

1. INTRODUCTION The interplay between engineering sciences, technology, and applied mathematics often leads to the analysis of initial and/or boundary value problems for nonlinear partial differential equations modelling real physical systems. Several interesting problems are also stated in terms of integrodifferential equations. As known [l], a large variety of solution methods can be developed towards the solution of the above class of problems. As a matter of fact, the selection of the proper method is one of the difficult task of applied mathematics and depends mainly on the structure of the problem to be solved and, in a minor extend, on the aims of the simulation. A standard solution technique of nonlinear initial-boundary value problem for partial differential equations is the collocation-interpolation method, see [2-41, originally proposed as differential quadrature method. If one consider the initial-boundary value problem for models described by partial differential equations, the application of the method goes through the following steps. (i) The space variables, say z, y, are discretized into a suitable number of collocation

points

Xi,YjPartially

supported

by MUBST,

Minister for University

Matematici nelle Science Applicnte.

13

and Technological

Research:

Project

Modelli e Metodi

N. BELLOMO

14

(ii) The dependent variable u = u(t, z, y) is approximated by the interpolant through values uij (t) = u(t, zi, yj) of the dependent variable in the collocation points. Typically, Lagrange polynomials and Sine functions are used for the interpolation. (iii) The space derivatives

are approximated

(iv) The initial-boundary nary differential Boundary

value problem

equations

conditions

ary of the domain

describing

being imposed of the independent

using the interpolation

is transformed the evolution in the collocation

mentioned

into an initial

of the values uij(t) points

in Item (ii).

value problem

for ordi-

of u in the nodes.

corresponding

to the bound-

variables.

(v) The solution of the initial-boundary value problem is then obtained solving the initial value problem mentioned in Item (iv) and interpolating, at each time step, the solution by the method This method model,

used in Item (ii).

discretizes

the original

with a finite number

of degrees

continuous

model (and problem)

of freedom,

into a discrete

while the initial-boundary

(in space)

value problem

is

transformed into an initial-value problem for ordinary differential equations. A similar method can be developed for the solution of mathematical problems for some classes of integro-differential equations. Again, the original problem is transformed into an initial value problem for the values of the dependent variable in the collocation nodes. This method is well documented in the literature on applied mathematics, it was proposed by [5] and stochastic Bellman and Casti [2] and developed by several authors in the deterministic framework [6,7]. As well documented, e.g., [S], it can provide a useful discretization of continuous models and efficiently deals with nonlinearities including the ones related to implicit boundary conditions. These features, make the method interesting for the solution of nonlinear problems in applied sciences. The survey [9] provides an interesting and detailed report on the application of the original differential quadrature method to several engineering problems. Additional references on applications can be recovered in several papers dedicated to this topic [8,10-161. On the other hand, it is known that the method does not generally work in some circumstances. For instance, referring to the Dirichlet problem, the classical Lagrange interpolation is not useful to deal with problems in unbounded domains and with solutions that are oscillating in the space variables. This problem was overcome by a suitable use of Sine functions, see [17-191, to the analysis of nonlinear problems [8,12]. A further important aspect is the estimate of the error bounds, generally developed by means of suitable comparisons between the computational solution delivered by differential quadrature methods and analytic solutions when they exist. This paper will show how further efforts can be developed towards improvements of differential quadrature methods. We may conclude that several studies have been developed to generalize and improve the mathematical method, which has been subsequently applied to interesting engineering problems. These improvements have generated a mathematical method, which will be called the generalized collocation methods, apparently useful to solve a large class of nonlinear problems in applied sciences. This review paper, which is the complement to the above cited paper [9], provides both an updating of the state of the art on this topic and a survey of some applications available in the literature. The contents are organized in three parts. The first part deals with the technical application of the methods. Here the application of the method is proposed already taking advantage of some recent developments. In particular, Section 2 provides technical information on the interpolation of functions and surfaces by collocation interpolation methods. Section 3 deals with the solution of initial boundary value problems for nonlinear partial differential equations in one space dimension both in bounded and unbounded domains. Section 4 is on the generalization of the method towards the solution of problems in two space dimensions.

Nonlinear Models and Problems

15

The second part of the paper is concerned with some technical developments of the method towards the solution of problems somewhat new with respect to the one reviewed in [9] and in Sections 3 and 4. Section 5 deals with decomposition of domains in one and two space variables also referred to as some ill-posed problems. Section 6 deals with the solution of integredifferential equations. Section 7 is about the analysis of some stochastic problems. The third part of the paper deals with the improvements of the method both towards develop ments of more efficient algorithms and the computation of error estimates. Section 8 deals with parallelization algorithms and error terms. Section developments

for initial boundary value problems and with the estimate of correction 9 deals with research perspectives towards further generalizations and

of the method.

This paper is proposed applications of generalized to be stated that classical methods

as a guide to applied collocation methods.

mathematicians interested in developments and In proposing the contents of this paper, it needs

in alternative to the above methods, nonlinear problems can be dealt with by of applied mathematics, e.g., finite elements [20,21], spectral methods [22,23),

splitting methods [24], TreEts methods [25,26], and so on. Indeed, the selection of one method with respect to others, can be done only for specific mathematical models and problems and is not object of speculations in this present paper. It is also mentioned that the problems dealt with in what follows are generally such that both independent and dependent variables are defined over a bounded domain and are subject to dimensionalization, with reference to the minimum and maximum values of each variable so that each one is defined over the interval [0, l]. For instance, if the space variables are defined over a rectangle [5,, z~ ] x (ym, ye], then the dependent variable u = u(t, 2, y) defines, after a suitable normalization of the dependent variables an application from [0, l] x [0, l] x [0, l] into [0,11.When problems are such that the dependent variables are unbounded, then further technicalities will be specifically indicated. Moreover, it is assumed, in the analysis developed in the sequel, that the solution exists unique and smooth in a suitable function space. Of course, this strong statement has to be properly verified for specific models and problems.

2. INTERPOLATION

TECHNIQUES

Consider the variable u = u(t, !c) defined over (0, l] x [0, 11, such that u is a one to one map from [0,1) into [0, l], for every t E [0, 11.Moreover, consider the collocation i=o,... which may be equally

,7L+1:13c={2~=o

)...(

2, )...,

X,+1=1},

(2.1)

spaced xi = ih,

h=I

(24

n+l’

or, for instance, Chebyschev-type collocation with decreasing values of the measure 1zi - xj( towards the borders, see Appendix I of [l]. In general, u = u(t,z) can be interpolated and approximated by means of by Lagrange polynomials as follows: n+l u(t, s) 2 uyt, x) = c Li(x)ui(t), (2.3) i=o or similarly,

by Sine-type

functions u(t, 2) e #(t,

where w(t) = u(t, zi) and where Lagrange by the following expressions:

nfl x) = c S,(x; h)q(t), i=o polynomials

and Sine functions

(2.4) are, respectively,

(x - x1). . . (x - X&1)(2 - Xi+1) *. . (x - 2,) x1). , . (Xi - X&1)(Xi - xi+l). . . (Xi - x72)’

Li(x)= (Xi -

given

(2.5)

N. BELLOMO

16

and sj(x.

The above defined interpolations

,

h)

=

sin(WW

- W)

(T/h)(X - ih)

P-6)

*

can be used to approximate

the partial derivatives of the

variable u in the nodal points of the discretization

(2.7) where &J(n)

= -&X”) drLh

(2.6)

af&l) = $$(Xi)

(2.9)

in the case of Lagrange polynomials, and

in the case of Sine functions. The value of the coefficients depends on the number of collocation points and on the type of collocation. Technical calculations provide, in the case of Lagrange polynomials, the following result: 1 l-k> &’ = c-, (2.10) zi - xh (zh - zi> n(zh) ’ h#i where (2.11) p#h

Pfi

Higher order coefficients may be computed exploiting the following recurrence formula: (r+l) _ - -

aji

c

(2.12)

h#i

Similar calculations developed in the case of Sine functions yield relatively simpler expressions, which are reported here for the first three derivative matrices (-l)i-j

aji(1) = h(i - j)' (2) = ‘ji (3) _ aji -

I& = 0,

2(-l)i-j+l h2(i - j)2

h3

’

---

a{?)=-1 n

If 2 3h’0

(2.13)

aii = 0.

In general, Sine-type interpolation requires equispaced collocations, while Chebyschev collocations can be used for Lagrange polynomials. Actually, the choice is not limited to the above interpolants. More in general, one can look for interpolations of the type n+l u(t,xc) 2

uyt, x) = c Xi(x)ui(t), i=o

Xi(G) = 1,

Xi(Xj) = 09

(2.14)

where x represents a general fundamental interpolant, such as Lagrange polynomials or Sine functions, but also fundamental splines, Legendre polynomials with Legendre collocations points, etc. The presentation that follows will, however refer, for simplicity, to the interpolations (2.3)

Nonlinear

and (2.4) leaving the technical

to the interested

results

reported

reader

Models

further

and Problems

17

generalizations,

of

in [27].

Moreover, it is well known that accuracy of the interpolation of the proper collocation points (to be related to the selection will be discussed Section 9 with reference to the computation the solution of problems. The same interpolation

which may take advantage

can be used for time dependent

can be obtained by the selection of the interpolants). This matter of the approximation of errors in

functions

in two space variables:

u =

u(t, 5, y) defined over [0, l] x [O, l] x [0, a], such that u(t; z, y) is a one to one map from (0,11x [0, a] into [0,11,for every t E [0,11.Consider, j=o

)...)

in addition

to the collocation

m+l:I,={ye=O

)...,

yj ,...,

IS, the following

one:

ym+r=l},

(2.15)

it follows that i=n+1m+1 '11= u(t, 2, Y) 2 Unm(t, 2, Y) =

or

c

c

i=o

j=O

n+l

u = u(t,

2,

y) 2

(‘2.16)

&(+j(Y)wj(t),

m+l

unm Ct,29 Y> = C C si(T

h)Sj(Y;

h)Uij(t),

(2.17)

i=o j=o

or even mixed type interpolations n+l

u = u(t,x,y)

2 Unm(t,2,y)

=

m+l

)J yd i=O

L,(z)S,(y;h)u,(t)

(2.18)

Si(Z;

(2.19)

j=O

or, symmetrically, n+l

u = u(t92, Y) s Unm(t,2, Y) =

j=o

The approximation of the space derivatives analogous to the ones of the one-dimensional

m+l

C C

h)Lj(Y)Uij(t).

j=o

in the collocation case

points

is obtained

by calculations

(2.20) h=O

while mixed type derivatives

k=O

are given by formula

of the type

(2.21) where the coefficients a and b are given by the expressions reported depend on the type of interpolation and number collocation points. Note that the interpolation, both in one and two space dimensions, nodal points l&(t) = un(t, Xi), %j(t) = @Yt, %!/j),

in equations is exactly

(2.10)-(2.13) satisfied

in the (2.22)

while partial derivatives are only approximated. When problems in two space dimensions are not defined on nonrectangular domains, then the interpolation method needs to be technically modified. For instance, problems in polar coordinates are dealt with in [9]. When the domain is convex with respect to both axes and regular, then the following interpolation can be applied: i=n+l

u = U(t, z, y) =

Pm* Ct9 5, Y>

=

x i=o

m(i)+1

C j=O

L(z)Lj(Y)Uij(t),

(2.23)

N.

18

BELLOMO

where the number of collocation points along the y-axis depends on the collocation on the z-axis. Alternatively, one can try to map the domain in a circle or square by suitable change of dependent coordinates. We do not discuss, This

important

at this stage, the error bounds

topic

initial-boundary

will be dealt

We summarize, partial

METHODS

in this section,

of initial-boundary

differential

dimensionless

in Section

of the interpolation-approximation 8 with

direct

reference

method.

to the solution

of

value problems.

3. SOLUTION the solution

with

equations

resealed

the application

described

in one space dimension.

variables

SPACE

DIMENSION

of the classical differential

value problems

quadrature

method

to

by second and higher order semilinear

In particular,

we consider

equations

with

such that u = u(t,x)

and the following

IN ONE

: [O,l] x [O, l] -+ [O, 11,

class of second order partial

differential

(3.1)

equations (3.2)

where q, p, and f are assumed to be given functions of their arguments and the perturbation parameter E is not necessarily small. As we shall see in what follows, this class of models already includes a large variety of interesting models of applied sciences. More general cases will be discussed later in this section. The content of the section is developed first by stating the two points Dirichlet, Neumann, and mixed problems. Then some generalizations, e.g., higher order problems, or problems in unbounded domain, will be developed. Bearing this in mind, consider the problem of computing the time-space evolution of the dependent variable u = u(t, z) in the following cases. PROBLEM (DIRICHLET) initial condition

3.1.

Consider

the initial-boundary

boundary

(3.2) with (3.3)

conditions u(t,O) = a(t)

where cp is a given smooth

function

and

u(t, 1) = p(t),

$(t,~)

= r(t)

where y and 5 are given smooth 3.3.

u(t, 0) = where (Y and S are given smooth gtt,

and

functions

Consider

a(t)

$$t,

(3.4)

and

functions and

functions

I) = b(t),

value problem

functions

for equation

of time. (3.2) with

vt E [O,ll,

(3.5)

of time.

equation

0) = -Y(t)

where 7 and p are given smooth

kft E ]O,ll,

of space, while Q and 0 are given smooth

PROBLEM (NEUMANN) 3.2. Consider the initial-boundary initial condition (3.3) and Neumann boundary conditions

PROBLEM (MIXED) boundary conditions

for equation

vz E IO, 11,

u(O, x) = cp(z), and Dirichlet

value problem

(3.2) joined

E(k

1) = V),

to the initial

vt E [O,I],

of time, or to the boundary u(t, 1) = p(t),

of time.

condition

(3.3),

to the

(3.6a)

conditions

vt E IO, 11,

(3.6b)

Nonlinear Models and Problems

The application

of the method

1. The space variable

for the solution

is discretized

‘u = ~(t, X) is interpolated

of Problem

into a suitable

and approximated

in equation (2.3) or equation same interpolation.

Boundary

conditions

being

3.1 goes through

collocation

imposed

by the values

in the nodal

the following steps.

Iz,, and the dependent

(2.4), and the space dependence

2. The initial-boundary value problem is transformed nary differential equations describing the evolution of the independent

19

ui(t)

= u(t, xi) as indicated

is approximated

points

on the boundary

of the domain

variables.

The above calculations yield a system of ordinary differential evolution of the variable u in the nodal points, that can be written

cp(Xi)+

+p(s,

t /( [

77(s, Xi) Hi(S)+

0

Xi)

K(s)

+ 2

=

apuj(s)

j=l

[

%+1(t)

using the

into an initial value problem for ordiof the values ~ij(t) of ‘u.in the nodes.

3. The solution of the initial-boundary value problem is then obtained solving value problem and interpolating the solution by the method used in Step 2.

z&(t) =

variable

the initial

equations defining the time in integral form as follows:

2 ajiuj(s) j=l

I

1( + Ef

s, xi, ui, Hi(S)

+ 2

aj$j(s)

j=l

)

0(t),

where Hi(s) = aoia(s) +u(,+~@(s) and Ki(s) = a$)a(s) +a~~\,$?(~), while the coefficients aij are given by the expressions reported in Section 2. The system can be solved by means of standard techniques for ordinary differential equations, see [l, Chapter 21. The solution of (3.7) linked to the interpolation (2.3) provides the solut*ion of Problem 3.1. The solution of Problem 3.2 can be developed analogously. The difference consists in the first and last equation which are obtained imposing Neumann boundary conditions. The result is the following: 1 uo(t) =

~ooqn+l)(Ta+l)

-

b(t)

-ao(n+l)

qn+l)oaO(n+l)

+ 2

%(n+l)wl

h=O

)I 7

t [ 2 1 J( cJjiUj(S) ds, s, xi, ui, Hi(S) + 2 +p(s, Xi) K(s) + 2 a~~4Q(s) + Ef 1( [ (3.8)

w(t) = cp(Xi>+

0

rl(s, Xi) K(S) +

ajiuj(s)

j=l

j=l

j=l

)I

N. BELLOMO

20

1 %+1(t)

=

~o(n+l)qn+l)o

-a00

6(t)

-

Mixed type problems conditions

(

(3.8 cont.)

.

h=O

I

with Dirichlet

on the other

1

ah(n+l)Uh

+ 2

(

qn+l)o

qn+l)(n+lpoo

boundary

side can be dealt

with

conditions

on one side and Neumann

following

the same

method

using

boundary the relative

expression given by equations (3.7) or (3.8) as first and last equations, respectively. The method has specifically referred to semilinear equations of the type given in equation (3.2). More in general,

one can deal with nonlinear

equations

of the type

;+(t,x,u,g ,...,g. The generalization

is immediate

(3.9)

using for higher order derivatives

the expressions

given in Sec-

tion 2. Similarly, one can deal with systems of partial differential equations. This generalization, that is also immediate, refers to the case where the dependent variable is a vector, say u. The solution technique leads to a system of equations for each component of u. Note that Sine interpolations are such that aii = 0. Therefore, the local coordinated does not contribute to space derivative. This feature has to be taken into account in dealing with Neumann boundary conditions, which can be obtained using splines for interpolations near boundaries. The solution method can be generalized to the analysis of problems in unbounded domains. Consider, in particular, the initial-boundary value problem for equation (3.2) in the half-space z E [0, co) with initial conditions p(z), and boundary conditions

40, x) = a(t), Rather compact,

e” - 1

x = log E

e”+l’ is such that gf

= 0.

(3.10)

than solving the problem by Scheme 1, simply putting U, = 0, it is convenient if possible, the whole half-space into the domain [0, 1) by the change of variable *=-

that

lim u(t,z) I-M

=;

z E [0, oo) + z E [0,1). The evolution

(1--22)0(t,$(Z))~

+/I(&$+))

(3.11)

= G(z), equation

[; (1 -*s)2z

to

can then be rewritten - iZ(l

-Z”)

as

21 (3.12)

+ Ef

.

t, $(z), ‘11,f (1 - z2) 2 >

The application of the method yields a system of ordinary differential equations for i = conditions. This method has the + 1 with ue(t) = o(t) and un+r = 0 as boundary 0 ,...,n advantage that equally spaced nodes for the variable L generates a discretization of the z variable such that i t =+ (xi - si-1) t, that is consistent for data decaying to zero at infinity. However, in order to optimize the choice of 1c,it is useful to know the type of decay at infinity.

4. PROBLEMS Consider

evolution

equations

IN TWO

SPACE

in two space dimensions,

that

DIMENSIONS may be written

as follows:

(4.1)

Nonlinear Models and Problems where u is the dependent

21

variable u = u(t, x, y) :

[O,11x [O,11x [WI + [--I, 11,

(4.2)

or, more in general u=u(t,x,y):[O,l]XD--+W, where the boundary

of D will be denoted

(4.3)

by aD.

Initial boundary value problems in two space dimensions to the ones used for the problems described in Section

can be solved by techniques analogous 3. In order to avoid repetitions, we will simply indicate how the interpolation can be organized, then a few guiding lines will be described in order to show the solution method exploiting the collocation interpolation described in Section 2. Again, we refer to a specific evolution that will be reported in Section 10. On the other

model having in mind the specific applications hand, the general evolution equation is of the

type %=f As in one-dimensional PROBLEM 4.1. initial

Consider

(t,X,Y,$$~,$,&

cases, we consider

,... ),

the following

the initial-boundary

value

(4.4)

problems.

problem

for equation

(4.1) or (4.4) with

condition u(0, x, Y) = cp(x, Y),

and Dirichlet

boundary

functions

Vx, y E dD : u(t;x,y)

consistent,

= a*(t),

for t = 0, with the initial

PROBLEM 4.2. Consider the initial-boundary value problem dition (4.5) and Neumann boundary conditions Vt E [O,l], given as smooth

functions

(4.5)

conditions

Vt E [O,l], given as smooth

VX,Y E D,

Vx, y E dD : n. Vu(t;x,

x, y E aD, condition

for equation

y) = y*(t),

of time, where n is the normal

(4.5). (4.1) with initial

z,y E dD,

to the boundary

(4.6)

con-

(4.7)

i3D.

Considering that the solution schemes are simply technical developments of already seen in Section 3, their presentation will be very concise. Consider first the solution of Problem 4.1. Then the interpolations defined in Section 2 yields a system of (n + 2) (n* + 2) ordinary differential equations corresponding to collocations (xi, yj) which can be easily written ss follows:

(4.8)

where partial derivatives are computed as indicated in (2.20)-(2.21). Moreover, the initial conditions are cp(xi, yj) and boundary conditions of the type (4.6) are imposed implementing the given time dependent values of the dependent variable Ujj(t) = Cl*(t), Similarly, one can deal with boundary conditions the space derivatives similarly to the one-dimensional

xi,yj

E

8D.

(4.9)

of the type (4.7) imposing compatibility of case. The schemes that we have just seen

N.

22

BELL~MO

can technically solve problems (4.1),(4.2). In principles, both interpolations, Sine or Lagrange, can be used. Considering that two interpolations are applied, one for each independent variable, then one can use, for each of them, different interpolations. Similarly, one can deal with systems of partial differential equations. This generalization, that is also immediate, refers to the case where the dependent variable is a vector, say u. The solution technique leads to a system of equations for each component of u. The method can be developed in a semiinfinite strip: z,y E HP+x [O,g] or z,y E [O,l] x HP+; or, still in two dimensions if both variables are defined in the half-space 5, y E HP+ x R+. In particular, in this case, the following change of variables can be applied: *=-

e” - 1

e”+l’

..J - ey - l eY + 1’

z = log s

= ?/J(z),

y = log g

= $W),

(4.10)

that is such that T, y E [O,00) + z, z’ E [0, 1).

5. DECOMPOSITION OF DOMAINS AND COLLOCATION METHODS As known, decomposition methods are developed either when the same algorithms or models are not used in different subdomains of the domain D of definition of the space variable. In this case, it may be convenient to decompose D into various subdomains, solve in each domain the initial-boundary value problem having properly matched, on the boundaries of each D,, the solutions in each subdomain. More in detail, the development of the method starts from the decomposition of D into a certain number of domains 0, D=

(jDv,

v=

l,...,N,

(5.1)

i=l

such that aL$,, is the boundary common to the domains D, and D,. The solution technique goes through the application of the following steps. (i) Discretization of the space variables into each subdom~n D,. (ii) Derivation of a system of ordinary differential equations for the variables 2~5 related to the variable u corresponding to the nodal points of each D,. (iii) Imposing the boundary conditions for all U; such that xi, yj E OD. (iv) Imposing the compatibility conditions u~j = u$, and if necessary analogous ones for the space derivatives, for all zi, yj E i?D,,,. (v) Solution of the system of ordinary differential equations that follows from the application of Steps (i)-(iv). P~ic~~ly interesting, in modelIing sciences, is the application of the method when different models are developed in each D,. This may occur also in physical situations such that the equation changes of type. Then the matching has to be organized having in mind both the different structure of the models and the algorithms used for the solution of the solution of the related evolution problem. This topic is introduced in [l, Chapter 31 with reference to mathematical problems related to partial differential equations. The literature on domain decomposition methods is v&e, often related to linear problems. Among several ones, we cite the analysis by Lybeck and Bowers [ZS]related to Galerkin methods and by various authors, as documented in 129-323, refated to the solution of problems in fluid dynamics by spectral methods. The survey by Quarteroni [33] provides a complete report on methods and literature on decomposition techniques related to spectral and finite differences methods.

Nonlinear Models and Problems Domain

decomposition

and unknown detail,

methods

source terms

the statement

have been applied

related

in [15,16) to identify

to heat diffustion

of two problems

proposed

23

models

boundary

conditions

in one space dimension.

More in

in [15,16] is the following.

PROBLEM 5.1. UNSPECIFIED BOUNDARY CONDITION. The initial boundary value (Dirichlet) problem related to equation (3.2) is stated with the proper initial conditions v(z) and with only one of the two boundary conditions, a(t) or p(t), needed for its solution. The solution to the initial-boundary-value

problem zl(t, z*) = u*(t)

is given Vt E (0, l] and z* ~]0,1[. condition

and the solution

The problem

consists

to the initial-boundary-value

PROBLEM 5.2. UNSPECIFIED SOURCE TERM. addition of a well-localized source/sink term

(5.2) in determining

the unknown

boundary

problem.

The mathematical

model is characterized

s(t) S(z - r,),

by the

(5.3)

which has to be added to the right-hand side term of (3.2). In equation (5.3), 6 is the Dirac delta function. The mathematical (Dirichlet) problem is stated with the proper initial and boundary conditions. The time-evolution of the source term s(t) is not known. The solution ?L(t, Z*) = u*(t)

(5.4)

to the initial-boundary-value problem is given Vt E [0, I] and for some z* ~]0,1[ . The problem consists in solving the initial-boundary-value problem and, in particular, in computing the source term s(t). The solution

of the above problems

is now well documented

in the literature

and reported

in [l,

Chapter 41. These problems are strongly motivated in technology as documented in the book [34] with reference to air conditioning problems, and to [35] with reference to heat control problems during machining. The analysis can also be generalized to problems with moving boundary as documented in (15). The whole matter is now object of a generalized analysis documented in [36], where the method is developed in a general framework based on sine-type interpolation and matching between different type of models based on their classical classification into parabolic, hyperbolic, and elliptic equations.

6. INTEGRO-DIFFERENTIAL

SYSTEMS

Several interesting problems in applied sciences are described by integro-differential equations. The collocation method can be developed towards the solution of problems related to the above mentioned class of models. This type of analysis is documented in [l, Chapter 41. Also, in this case, we will refer to a specific class of models, rather than an abstract class of equations, to show the practical application of the method to be followed by its generalizations. In details, we deal with following class of models:

($ +u(t,u)$>

f(C4 =

=

WI

JJ~(21,w)tilr(v, ‘1~)f(t,v)f(t>w) w;

D D

where u,v,w E D. As a particular case, it can be recognized

Jllcl( 0

then equation

v,w,u)du

that

= 1,

dv dw - f(t,

J

u>D rl(u,w)f(t, w>dw,

03.1)

if D = [0, 11, a = 0, and Vv,w

E [O,i] x

(6.1) is the Jager and Segel model proposed

in [37].

10,11,

(6.2)

24

N. BELLOMO

This model is such that u has the physical meaning of state variable for an individual of a large population of anonymous interacting subjects. ~(v, w) is the encounter rate between individuals with state v and w, respectively, and $J(v, w; ?J) is the probability density that after the encounter between individuals with state v and w, the individual with state v ends up into G. Under suitable integrability properties, the solution satisfies the following: d ’ f(t, u) du, = 0, Vt 2 0. (6.3) 2, J We shall first refer to this particular model with interest in solving the initial value problem

$@, u) = J~fl(o-4,

(6.4) fo(u) = f(O,u), where J[f] represents the right end side term of equation (6.1). Existence of solutions was developed in [38] with reference to model proposed in 1371and its generalizations. The application of the method will now be shown with reference to problem (6.4). Then some technical developments are discussed. Consider then the collocation of the u-variable i= 0 ,...) m:1,={211=0 and the approximation

).‘. ,21i).,.,. 21m=l),

(6.5)

of f

f(G ‘ZL) g f”(& u) = 2 xi(u)fi(t), (6.6) i=l where f+(t) = f(t?ui) and xi denotes the i-order fundamental polynomial, may be Lagrange or Sine interpolant. Then following the line of the preceding sections, it is immediate to obtain a system of ordinary differential equations

$%I=fyF Gpr?fp(U*(t)- fiw 2 4Pfrm

(6.7)

Gip.2= I’

W)

p=o q-o

p=o

where I’ 17(v,wM( v, w; ~i)xp(v)x&4

and

dv dw,

1 i$, =

q(ui, w}~p(w) dw. (6.9) J0 The generalization to class of equations different from the one we have just examined is almost immediate although several computational difficulties may arise and have to be technically solved. For instance, some models in mathematical physics are stated in terms of integredifferential equations such that the state variable is defined over the whole real axis, 21 E W or over the positive real axis u E lR+. Examples are the models of nonlinear kinetic theory [39]. In this case, u is a vector, The variable u in some models of the kinetic theory is such that some of the components of u are defined on a bounded domain and some on the whole real axis. This is the case, for instance of the Boltzm~n equation in polar coordinates in the velocity variable [401. Then, mixed type interpolation can be useful. The analysis of this section is limited to the initial value problem. The physical interest of this problem refers to systems that are homogeneous in the space variable. In principles, one can develop a similar analysis for integro-differential equations with partial derivative with respect to the space variable. Again this is the case of the models of kinetic theory. However, as well documented in 1391, the treatment of boundary conditions for this class of equations involves severaf additional technical problems that are not dealt with in this paper. On the other hand, enforcing boundary conditions to equation (6.1) in ?_I. = 0 or % = 1 is a relatively simpler problem. In this case, boundary conditions have to be related to the expression of the term Q and to suitable conservation hypothesis. An example of statement of boundary conditions related to conservation of mass can be recovered in [41].

Nonlinear Modelsand Problems

7. STOCHASTIC

25

PROBLEMS

Stochasticity may be an important feature of mathematical models and problems in applied sciences. Indeed, a real system is never fully isolated from the outer environment. If one models, in a simple way, the actions of the outer environment over the inner system, it is quite natural to do it in a stochastic manner. In other words, considering that one generally does not describe both inner and outer systems, a reasonable way to model the whole system is to assume that the outer environment stochastically perturbs the behavior of the inner system. Uncertainty may also be related to the formulation of the mathematical problems. The physical situation, may be such that initial and/or boundary conditions can be known only with uncertainties and may need to be stochastically modelled. Interesting models described by stochastic partial differential equations can be found both in solid and fluid mechanics as documented in the paper by Markov (421 and in the book by Capinski and Cutland [43]. Further, we recall, with reference to [15,16], that inverse-type problems, which may arise in inverse-type problems related to the analysis of mathematical models in applied sciences. These problems are such that some information needed by the statement of the mathematical problem is not given and is substituted by suitable information on its solution. The information on the solution must be exploited to determine completely the model and/or the problem. Several motivations suggest that a large part of inverse problems should be organized in stochastic rather than deterministic terms. In indeed, the information on the solution to the mathematical problem is obtained by direct measurement of the real system. Consequently, stochasticity may be generated either by the inaccuracies of the measuring instruments or by the noise induced by the direct influence of the outer environment on the inner system. Moreover, generally the real system involves a number of variables much higher than those taken into account in the mathematical model. Therefore, the measured quantities may fluctuate because of the influence of the neglected variables. The modelling of the stochasticity is achieved by the superposition of a stochastic perturbation, a noise in time or space, to the deterministic values. One of the interesting generalizations of differential quadrature methods refers to the solution of stochastic problems as documented by paper [6], and by book [7]. In dealing with stochastic problems, one has to tackle additional difficulties with respect to deterministic analysis. In particular, one has to deal with the representation of the solution by linking to suitable collocations both in time and space the probability distribution over the values of the dependent variable corresponding to the above collocation points. As we shall see, generalized collocation methods can be developed to obtain the aforementioned probability distributions. Bearing this in mind, some additional notations need to be stated. In particular, consider a complete probability space which is denoted by the triple (a, .F, P), where 0 R is a nonempty set of elementary events; l F is a a-algebra of subsets of R; l P is a probability measure on (Q, 7). Fieferring now to (0, F, P), the following definitions can be given. DEFINITION 7.1. STOCHASTICPROCESS. A function u defined on [O,T] x 0, where [O,T] is a time interval,is a stochastic process if, for any t E (O,T], u(t, .) : R t-+ W is a random variable. DEFINITION 7.2. REALIZATION AND PATH. Let t E [O,T] be fixed, then the random variable u(t;w) is called the realization of u at time t. Moreover, for fixed w E R, the function [0, T] u(t; w) is called the path of u. DEFINITION 7.3. TIME-DEPENDENTSTOCHASTICFIELD. A function u defined on [0, T] x D x R is a time-dependent stochastic field if, for any t E [0, T] and z E D, u(t, 2, .) : 0 W is a random variable.

26

N. BELLOMO

DEFINITION 7.4. PROBABILITY. The probability, denoted by Fl(U; t, x), that the random able u(t, x, w) at time t and in x, takes a value not grater than U is FI(U;t,x)=P(wER,u(t,x,W)
v-a&

(7.1)

where U is a real number. The following cases of stochastic problems can be listed as follows.

6)

Mathematical models with parameters known as random variables: r = r(w) : R -

IR.

(7.2)

(ii) Mathematical models with stochastic parameters known either as time stochastic processes r = r(w, t) : 0 x [0, T] -

IR

(7.3)

or as space stochastic processes r=r(w,x):Rx(O,l]-R.

(7.4)

(iii) Mathematical models with additional weighted noise. In one space dimension models of the type $

=f(t,s,u,g

,...,

g;r(t,x))

(7.5)

+io(t,x,u)$,

where w is the noise and $ is its “formal derivative”. (iv) Mathematical problems with deterministic boundary conditions and random initial condition ug = UO(W,x). Mathematical problems with deterministic initial condition and stochastic boundary con(v) ditions known as stochastic processes. In all cases, the solution to the initial-boundary chastic

value problems is a time-dependent

sto-

field ‘1L=‘1L(W,t,x):52x[O,T]x[O,1]-IW.

(7.6)

The objective of mathematical methods is to provide the representation of such a field. Therefore, it is necessary to provide some more information on the concept of probability density. In detail, if we consider a discretization of the time interval [0, T], then the set of random variables u(w) = {W(~),...,%(W))

(7.7)

can be defined for each point of the time discretization at fixed x, where Us In this case, the n-probability is denoted by Fn(&,...,Q;tl,

. . ..L.x),

which defines the joint probability that u~liI Vi for all i = 1,. . . , n. We restrict our attention to the case where FI or F, has a probability or f,,, which is defined by

J

(I

--oo

and

--oo

(7.8)

density,

respectively, fi

fi (a; t, z>& = FI (U; t, xl ,

Ul J ...JU”f?l(U1,...4n;tl,...,L,x) --oo

= u(ti, w).

du1-..du, = F,(Ul,...,U,;tl,

(7.9)

. . ..&.x).

(7.10)

Nonlinear

Models

and Problems

27

If fi is known, then the first-order statistics, i.e., all moments of u, can be computed. Similarly, the n-order statistics, which involves all moments and correlation of the vector u can be developed if fn is known. In analogous fashion, one can discretize also the space variable and reach the n x m probability density (7.11)

fnm(~11,...,~,1,...,~1m,...,~~m;~lr...,~n,~1,...,~,),

where ~ij = u(ti, sj). The selection of the order of statistics depends on the level of accuracy which is required to the mathematical model. In several cases, it is sufficient to deal with the first probability density distribution

fi(~; t, z).

Considering that a time dependent random field is known if its n x m probability

density is

known for all type of discretization including boundary and limit conditions, then the aim of the stochastic calculus related to the solution of problems refers to computing such a density and provide a suitable representation of the statistical measures. Reference is made to paper [13]. Consider first the case of a discrete scalar system and assume that the first probability density can be computed. Then, the first-order statistics can be organized. In fact, the knowledge of f leads to the computation of all moments of u at fixed time and space. In particular, one has the p-order moments and central porder moments

I

E{uP}(t, z) =

upfl(u; t, z) du

(7.12)

DU

and &{uP)(t, respectively. The second-order in time and space fz

statistics

(‘1L(hr 21, 4t2,

z) = s,

”

Iu - %)lP

(7.13)

fr(u; t, z) du,

require the knowledge of the second-order probability

z>; t1, t2, x)

>

f2 (46

a),

4k

x2); t, Xl, x2)

The knowledge of the densities defined above leads to the computation time and space. For instance, the autocorrelation in time or space R (ti, tz; z> = E {u (tl; ~1 ‘1~(t2; ~1) ,

R (t; x1,22) = E {u (t; 4

density

(‘7.14)

.

of the correlations

in

‘1~(t; 22)).

(‘7.15)

x)))

(7.16)

In a similar way, one can compute the autocovariances C (tl, tz; z) = E {(u(ti;

z) - -Wu}(ti;

z)) (4t2;

x) -

E{u)(t2;

and C (t; 51, ~2) = E {W;

~1)

-

E{u)(t;

21))

(4t;

~2)

-

E{uHt;

x2)>)

”

(7.17)

Similar representations can be developed for higher order statistics. The solution of initial-boundary value problems can be obtained by differential quadrature methods only in some relatively simpler problems, that is, when the mathematical modelling of the randomness are such that the trajectories Ui=‘U.(Wi;t,5),

Uj=7L(bJj;t,X),

wi,wj

E

R

(7.18)

are uncorrelated for all wi,wj E R. In the case of uncorrelated trajectories, sample solutions are possible. Sample solutions lead to the identification of random variables by their moments. Further, the analytic representation of the probability density can be obtained by orthogonal expansions with coefficients related to the moments. This method is documented in [44] as well as in [l, Chapter 41.

N. BELLOMO

28

8. PARALLELIZATION ALGORITHMS AND ERROR BOUNDS One of the main withdraws of the application of the method described in this paper is related to the fact that accuracy cannot be obtained increasing indefinitively the number of collocation points. This certainly improve the evaluation of the space derivatives in the collocation points. On the other hand, when the number of equations increases, the system of ordinary differenti~ equations generates further computational errors related to the large dimension of the system. In general, one should expect that by increasing the number of nodes, the gap between the true solution and the one obtained by the application of the method will gradually decrease up to a certain value nC. On the other hand, for n > n,, such a gap generally increases. All this matter needs to be put in a more rigorous framework, that will be done later in this section. However, a paralielization method was proposed in [14] in order to overcome the above difficulty. The sequential steps of application of the method are the following. STEP

1. As first step, the collocation 1, is split into two subcollo~atio~ h=O,...

,p+1:1~=(2*=O,...*z~

k=O,...

,q+l:Ig={za=O

,*.., Icp.+P1)

(8.1)

and ,...) Zk )..., z,+l=l},

(8.2)

such that n=p+q,

11 U&

= 1,.

(8.3)

STEP 2. Set the initial value problems correspondi~ to both collocaGons in a way that initial conditions are set at t = to and boundary conditions are enforced for both systems for h = k = 0 and h= (p+l), k(q+l).

both systems from to to tl = to + A using for both of them a third-order algorithm

STEP 3. Integrate

Runge-Kutta

w&l = w&o + 46 (Kw -I- 4%/zo(K10) + &o@Go,

UM = URO+

$ (Go -t 4&0(&o)

&ho))

t

+ K~{K~o,K~&o)),

V-5)

where K1 is computed using all points of the collocation. STEP 4. Once u1 has been computed, successive integration steps.

the procedure is repeated starting from Step 2 for all

The procedure is essentially the same for systems of equations. Simply, each subcollocation generates a system of ordinary differential equations corresponding to each partial differential equation. The method can be developed also using more than two sub~ollo~ations. Step 1 is modified and consequently in Step 2, one has a number of systems corresponding to the number of colloca;tions. Then the stiffness of the systems of ordinary differential equations is further reduced. Of course the method can be developed also using more accurate integration algorithms, e.g., higher-order Runge-Kutta methods. Step 3 is technically modified in order to deal with different algorithms. The above parallelization algorithm can improve accuracy of the computation, as shown in [14). However, the computation of the error bounds needs to be carefully studied, not only because it gives a direct information on the level of accuracy of the solution delivered by the mathematical method, but also as it may provide the necessary info~ation to reduce the number of collocation points, and hence, the computations time necessary to obtain the required accuracy. In general, the error is propagated by the fact that the interpolations are exactly satisfied in the collocation points, but they may be unprecise in the intervals between the collocation points.

Nonlinear Models and Problems

Moreover,

the estimate

of the space derivatives

this type of local error generate by the collocation be in to all

is not exact even in the collocation

the gap between

the true and the approximated

points.

solution

Thus,

delivered

method.

Before continuing analysis will Bearing this with respect to space for

29

on this matter,

developed, also for mind, consider the time for all values values of the time

we need to state

more precisely

the concept

of error.

The

simplicity of notations, for problems in one space dimension. space of the functions u = ~(t, z) with bounded pderivative of the space variable and bounded q-derivative with respect variable. Let un = un(t,z) be the solution delivered by the

collocation method and u = u(t,z), the one of the initial-boundary value problem. that u, U” belong to a normed space of smooth functions, then the following definition

Suppose of error

can be proposed: en = lllu - unIIl = s;~l

(8.6)

et(r),

where e;(t)

= llu - u”ll(t)

= J$El

Iu - d(t).

(8.7)

Actually, a direct estimate of et, and hence of e, can be given only in the case of problems with known analytic solution, as also shown in Lecture 4. Although these problems are often a useful test as documented in [9], problems of interest in applied sciences very unlikely are characterized by analytic solution. In general, one has to start from the error in approximating, at t = 0, the space derivatives. Then, the estimate of the error propagation can be obtained according to Gronwall’s lemma [45]. Several particular,

problems addressed to improve accuracy of the solution the following aspects needs to be developed.

need to be dealt

with.

In

Development of best approximation method of the initial datum in order to make as small as possible the initial error EL. Development of best interpolation method along the time integration in order to make as small as possible the error ey. Selection of the number n of collocation points and of the time step of the integration in time. Improvement of the algorithms for the integration in time. Development of mathematical methods for the estimates of the error bounds. The crucial point dependant functions such that

in the application of the collocation method is the approximation of time starting from t = 0. Referring to EL, one has to develop approximations lim E: = 0, Tl-NX

(8.8)

and n>n*,m>m**s~
(8.9)

Polynomial (Lagrange) approximations satisfy the above conditions cation is applied. The second condition is often satisfied although general rule.

only if a Chebyscheff cello it cannot be regarded as a

Certainly, a good approximation of the initial condition, and subsequently of time dependent conditions is the starting point to improve accuracy of generalized collocation methods. However, increasing indefinitively the number of collocation points does not imply that e; and e” will decrease. Indeed, when the number of differential equations become large, integration errors may not be controlled even by reducing the time integration step. The application of generalized collocation the following, among several ones, items:

methods

need to go through

the proper

analysis

of

30

N. BELLOMO l l l

selection of the type of collocation and of the interpolants; selection of the number n of collocation points; selection of the time step h of the integration in time.

The first item was already discussed in the introduction. The second and third items, which are somehow related each other, may be quite difficult, or even impossible to be solved theoretically. Therefore, we suggest a computational experimental analysis which may provide useful indications on the above problem. The method is developed through the following steps.

0)

Let n be fixed, then h can be decreased by a step Ah until the distance in norm (8.10)

dh = 111%- Uh-Ahll( tends to a sufficiently small value. The optimum value of h, obtained procedure, corresponding to a certain n is denoted by h,. (ii) Consider now the distance in norm d, =

Ilbn - ~n+~lll,

h=

h,,,

by the above

(8.11)

and increase n until the distance (8.11) tends to a sufficiently small value. Actually, dn versus n first decreases (not necessarily with monotonic behaviour) until n reaches a critical value n,. Then for n > nc, the distance d, may start to increase. This means that further increasing of n for n > nc does not give any practical advantage. Thus, considering that the corresponding value of h, increases more than linearly, one should consider n, the optimum value of n to be selected for practical calculations. It is plain that this procedure is an experimental one which does not give any quantitative estimate of the error bounds, but simply represents a practical and generally efficient way for a proper selection of h and n. Further work has to be done towards the estimate of the error bounds. Let un be the solution to the initial boundary value problem obtained by the application of the collocation method. The true solution u = u(t, 2) may be related to rP(t, z), which we know how to compute, and to an error r(t, z), which one wishes to estimate. Accordingly, one can write u(t, x) =

uyt, x)

+ r(t, x).

(8.12)

Substituting the above expression into evolution equations such as the semilinear equation (4.1), yields in a large variety of cases, e.g., polynomial nonlinearities, the following system of two coupled equations:

sun au” -at = dt, x)x + P(C

4% +ef (,,x.u~,~)7

dr - = q(t,x$

+pp,xg

at

+ef

(

t,x,zP,r,-

dun

dr

-

ax 1ax >

(8.13)

.

The above system has to be solved with the following initial conditions:

UV,

xl, xc>,r@,

vx E [O,11,

(8.14)

and boundary conditions qt,

o>, w,

11,

r(O,x)

= r(t, 1) = 0,

vt E [O,l].

(8.15)

If one develops a method to obtain ~~(0, x) and r(0, x) from the information on q(x), then the collocation method can be applied for the solution to the above initial boundary value problem to obtain an approximated calculation of the correction term.

Nonlinear Models and Problems

This information tions of the initial

also gives indication condition,

on the stability

being plain that:

31

of the system

with respect

to perturba-

~(0, Z) = 0 =S r(t, X) = 0.

Therefore, we have to indicate some reasonable criteria ~(0, X) # 0. The following two methods are suggested.

to evaluate

a suitable

expression

METHOD 8.1. Let n and h be fixed. Then the initial value problem is solved for t E [0, h] with a number m of collocation points larger than n and a time-integration step smaller than h. ‘This should provide, for t = h a better approximation than the one obtained with n, h. Then the

choice is r(h,z) METHOD

8.2.

Let the initial

condition

= P(h,z)

- tP(h,x).

(8.16)

for un be given as follows: n+1

‘1L”(O, x) =

c cpiXi(X),

(8.17)

i=o and let

m+l u”(O,2)

=

c

(8.18)

‘piXi(Z)

i=o be a relatively better approximation obtained by a collocation poins. Then the choice is r(0, X) = P(0, X) - ?P(O, Z). In both cases, the choice does not provide an exact estimate useful one both for stability analysis and error development.

with a larger

number

of nodal (8.19)

of the initial

error,

but simply

a

9. DEVELOPMENTS We have seen that the differential quadrature method can be efficiently applied to the solution of several interesting problems in applied sciences. The method, as already mentioned was proposed in the paper by Bellman, Casti and Kashef [2,3] and developed and improved by various authors. This survey paper is concluded by this section which proposed some research perspectives towards further development and improvement of the method. The contents of this section is naturally developed according to the authors’ bias. Therefore, we will concentrate, among several ones, on the following research perspectives. In detail, the following two topics will be dealt with: solution problems with nonlinear boundary conditions, and improvements of the interpolation techniques. Nonlinear

Boundary

Conditions

The statement of linear Dirichlet or Neumann boundary conditions, in applied sciences, may be an idealization of physical reality. Boundary conditions may be given, in practical problems, by suitable compatibility conditions on the boundary of the domain of definition of the space variables. Then one has nonlinear algebraic or differential boundary conditions as documented in [46]. Consider then the initial-boundary value problem for scalar equations in one space dimension with boundary conditions stated as follows: x = 0,

vt E (0,11: WI =

x =

vt E [O,ll

1,

where Q, 0, a, and b are smooth functions consistent with the initial condition

: %+1

Q(‘ILo, %+1)

+ a(t),

= @(‘ILo, %+1)

of their arguments.

Boundary

~(0) = 4~0(0),~~+1(0)) + 4% ~(1) = P(~o(o),‘~~n+l(O)) + b(O).

(9.1)

+ b(t),

conditions

need to be

(9.2)

32

N. BELL~MO

Then, the evolution problem is solved recovering the boundary conditions for the discretized system at each step of the time integration t,. When it is possible to obtain their value at the proceeding time t,_ 1, one has

w(tn) = ~(~O(tn-l),~,+t(t,-l)) + ~&-I>, ~~+~(t~)= ~(~o~~~-l~,~~+I~tn-l~f +%-I).

(9.3)

In general, it is useful to obtain from equations (9.1), an evolution equation for the dependent variable on the boundary to be solved by implicit or explicit methods. Improvements

of the Interpolation

‘Bchniques

The application of the method is based on the use of fundamental interpolants. Typically Mixed type interpolations can be used for problems Lagrange polynomials or Sine functions. in two space dimensions. We have also discussed, how one can improve accuracy by linking Lagrange-type interpolations to the proper Chebyceff collocation. A general rule, to be however practically tested, is that one can use Sine functions for oscillating in space solutions and Lagrange interpolations for nonoscillating solutions. However, it is reasonable to look for improved interpolation techniques. In this line one looks for spectral approximations rather than interpolations. The representation of a function u = u&z) is then given by an expansion of the type

where the functions I+!Q belong to a complete space of orthonormal functions defined in a suitable Hilbert space with weighted (the weight function is denoted by w = U(S)) inner product

(f,

=

9)&)

Jfk

z)9&

~)W(Z) dx*

(9.5)

Consequently, the coefficients q are given by: q(t) = (u, ~~)~, where the calculation of the integrals may need numerical computation. If u is approximated by 21”corresponding to the collocation &, then a truncated expansion is used i=n

u(t,

2)

=

zP(t,2)

=

c

i=-* where the coefficients ci have to be computed collocation points

ci(t)?+&(z),

(9.6)

numerically exploiting the values of = in the

(9.7) where IV’ denote the weights in the quadrature formula. F&placing the above expression in the the evolution equation, e.g., equation (4.1), yields

(9.3)

Nonlinear

Taking

the inner

product

yields

Models and Problems

an evolution

equation

33

for the coefficients

of the truncated

expansion

(9.9) . UJ This procedure may be technically more elaborate than the direct application of collocation methods. However, may allow to select more efficient polynomial approximation such as wavelets, see [47,48], and recent

developments

[49,50].

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