Camargo et al. Boundary Value Problems 2013, 2013:45 http://www.boundaryvalueproblems.com/content/2013/1/45 R E S E A R C H Open Access One-sided and two-sided Green’s functions Rubens de Figueiredo Camargo1*, Ary Orozimbo Chiacchio2 and Edmundo Capelas de Oliveira2 *Correspondence: rubens@fc.unesp.br 1Departamento de Matemática, Faculdade de Ciências, Unesp, Bauru, SP 17033-369, Brazil Full list of author information is available at the end of the article Abstract We discuss the one-sided Green’s function, associated with an initial value problem and the two-sided Green’s function related to a boundary value problem. We present a specific calculation associated with a differential equation with constant coefficients. For both problems, we also present the Laplace integral transform as another methodology to calculate these Green’s functions and conclude which is the most convenient one. An incursion in the so-called fractional Green’s function is also presented. As an example, we discuss the isotropic harmonic oscillator. 1 Introduction There are several methods to discuss a second-order linear partial differential equation. Among them we mention the simplest one, the method of separation of variables, and the method of integral transforms, particularly the Laplace transform, which is in many cases most convenient []. On the other hand, after the method of separation of variables, we get the following general second-order linear ordinary differential equation: a(x) d dx y(x) + a(x) d dx y(x) + a(x)y(x) =  on the interval a < x < b, whose corresponding nonhomogeneous one is given by a(x) d dx y(x) + a(x) d dx y(x) + a(x)y(x) = F(x), where F(x) is a forcing term. Assuming that a(x) is a continuously differentiable positive function on this interval and that a(x) and a(x) are continuous functions, we can write the above ordinary differential equation as follows: d dx [ p(x) d dx y(x) ] + q(x)y(x) = f (x), () the so-called self-adjoint form known also as an ordinary differential equation in the Sturm-Liouville form. In this equation, p(x) and q(x) are continuous functions that are related to the coefficients a(x), a(x) and a(x). The nonhomogeneous term, f (x), is also related to F(x) []. The methodology of the Laplace integral transform is adequate to discuss the one-sided Green’s functiona because the initial conditions, in general, are given in terms of the own function and the first derivative. A simple question arises when we discuss the two-sided Green’s function associated with a problem involving boundary conditions, i.e., is the © 2013 Camargo et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.boundaryvalueproblems.com/content/2013/1/45 mailto:rubens@fc.unesp.br http://creativecommons.org/licenses/by/2.0 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 2 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 Laplace transform methodology convenient to discuss this problem? The answer depends on the sort of problem we are studying, as we will see in the following sections. On the other hand, the fractional harmonic oscillator was discussed in a series of papers by Narahari et al. [–] where they presented the dynamic of the fractional harmonic os- cillator, including also the damping, and by Tofighi [] who discusses the intrinsic damp- ing. In this paper we discuss Eq. () associated with an initial value problem and a boundary value problem. In both cases, we present two methodologies, the Laplace integral trans- form and the Green’s function methodology. After that we conclude which methodology is the most convenient one. We sum the paper up presenting the corresponding fractional case where we discuss the Green’s function associated with the fractional harmonic oscil- lator. Finally, we present our concluding remarks. 2 One-sided Green’s function To solve the self-adjoint differential equation, we introduce the so-called one-sided Green’s function, also called influence function, a two-parameter function, denoted by Gi(x|ξ ), which describes the influence of a disturbance (also known as impulse) on the value of y(x) at a point x, concentrated at the point ξ . If we know the Green’s function, as we construct below, the solution of an initial value problem composed of the self-adjoint differential equation and the initial conditions can be written as follows: y(x) = ∫ x a Gi(x|ξ )f (ξ ) dξ , () where y(a) =  = y′(a). For a fixed value of ξ , the one-sided Green’s function is the solution of the correspond- ing homogeneous initial value problem, i.e., for x > ξ we have the homogeneous ordinary differential equation d dx [ p(x) d dx Gi(x|ξ ) ] + q(x)Gi(x|ξ ) =  and the initial conditions Gi(ξ |ξ ) =  and d dx Gi(x|ξ ) ∣∣∣ x=ξ =  p(ξ ) . 2.1 Constant coefficients As a particular case, we consider the problem involving an ordinary differential equation with constant coefficients, which can represent a problem related to the classical harmonic oscillator and transmission lines, for example, d dx y(x) + b d dx y(x) + ( a + b)y(x) = f (x), () where a and b are positive constants, and the homogeneous initial conditions are given by y() =  = y′(). http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 3 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 First of all, we write the ordinary differential equation in the corresponding self-adjoint ordinary differential equation d dx [ ebx d dx y(x) ] + ebx(a + b)y(x) = ebxf (x), where we identify p(x) ≡ exp(bx). Thus, the one-sided Green’s function, denoted by Gi(x|ξ ), satisfies the following homo- geneous ordinary differential equation for ξ fixed: d dx [ ebx d dx Gi(x|ξ ) ] + ebx(a + b)Gi(x|ξ ) =  and the initial conditions Gi(ξ |ξ ) =  and d dx Gi(x|ξ ) ∣∣∣ x=ξ = exp(–bξ ). Two linearly independent solutions of the homogeneous ordinary differential equation are given by y(x) = e–bx a sin ax and y(x) = e–bx a cos ax which furnishes for the general solution (the Green’s function) Gi(x|ξ ) = A(ξ ) a e–bx sin ax + B(ξ ) a e–bx cos ax, where A(ξ ) and B(ξ ) must be calculated by means of the initial conditions, i.e., by the following system: A(ξ ) sin aξ + B(ξ ) cos aξ = ,( – b a sin aξ + cos aξ ) A(ξ ) + ( – b a cos aξ – sin aξ ) B(ξ ) = exp(–bξ ). Solving the system above, we obtain A(ξ ) = cos aξ exp(–bξ ), –B(ξ ) = sin aξ exp(–bξ ) which furnish for the one-sided Green’s function, for x > ξ , Gi(x|ξ ) = exp[–b(x + ξ )] a sin [ a(x – ξ ) ] () and, as we know, satisfies the property Gi(x|ξ ) = –Gi(ξ |x). Finally, the solution of our initial value problem is given by y(x) =  a ∫ x  e–b(x–ξ ) sin [ a(x – ξ ) ] f (ξ ) dξ . () http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 4 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 2.2 Laplace transform Another way to calculate this one-sided Green’s function is by means of the methodology of Laplace transform. Multiplying Eq. () by the kernel of Laplace transform, exp(–sx) with Re(s) >  and integrating from zero to infinity, we have sF(s) – sy() – y′() + b [ sF(s) – y() ] + ( a + b)F(s) = G(s), where F(s) = ∫ ∞  e–sxy(x) dx and G(s) = ∫ ∞  e–sxf (x) dx. Using the initial conditions, we obtain an algebraic equation whose solution is given by F(s) = G(s) (s + b) + a . By means of the convolution product and the relation involving the inverse Laplace trans- form L – [  (s + b) + a ] = exp(–bx) a sin ax, we get y(x) =  a ∫ x  e–b(x–ξ ) sin [ a(x – ξ ) ] f (ξ ) dξ which is exactly Eq. (). 3 Two-sided Green’s function In the case that we have a two-point boundary value problem, i.e., when the boundary conditions are fixed on the extremes of the interval, we have a two-sided Green’s function, also called Green’s function, only. We consider the second-order ordinary differential equation in the Sturm-Liouville form d dx [ p(x) d dx y(x) ] + q(x)y(x) = –f (x) for x < x < x and the boundary conditions y(x) =  = y(x), and comparing this problem with the corresponding one-sided Green’s function, we have put –f (x) in the place of f (x) for convenience only. The two-sided Green’s function, denoted by Gb(x|ξ ) for each ξ , satisfies the following problem consisting of homogeneous ordinary differential equation d dx [ p(x) d dx Gb(x|ξ ) ] + q(x)Gb(x|ξ ) =  for x �= ξ and the homogeneous boundary conditions Gb(x|ξ ) =  = Gb(x|ξ ). http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 5 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 This Green’s function must satisfy also the continuity Gb ( ξ+|ξ) = Gb ( ξ–|ξ) and a jump discontinuity on the first derivative [] d dx Gb(x|ξ ) ∣∣∣ x=ξ+ – d dx Gb(x|ξ ) ∣∣∣ x=ξ– = –  p(ξ ) · The solution y(x) can be interrelated as a displacement and be given by y(x) = ∫ x x Gb(x|ξ )f (ξ ) dξ , where f (ξ ) is a force per unit length. Gb(x|ξ ) is the displacement at x due to a force of unit magnitude concentrated at ξ . In this case, we have Gb(x|ξ ) = Gb(ξ |x), the so-called reciprocity law. 3.1 Constant coefficients As an example, we discuss the following ordinary differential equation: d dx y(x) + b d dx y(x) + ( a + b)y(x) = –f (x) with the homogeneous boundary conditions y(x) =  = y(x). First, we construct the corresponding Green’s function, as we have seen before. Two linearly independent solutions of the homogeneous ordinary differential equation are y(x) = e–bx a sin ax and y(x) = e–bx a cos ax. Imposing that y(x) satisfies y(x) = , we can write y(x) = A(ξ ) a e–bx sin [ a(x – x) ] . On the other hand, to satisfy y(x) = , we put y(x) = B(ξ ) a e–bx sin [ a(x – x) ] , where A(ξ ) and B(ξ ) must be determined. Using the continuity at x = ξ and the jump discontinuity of the first derivative, we obtain a system of two algebraic equations involving A(ξ ) and B(ξ ) whose solution is A(ξ ) = sin[a(x – ξ )] sin[a(x – x)] and B(ξ ) = – sin[a(ξ – x)] sin[a(x – x)] · Thus, the Green’s function is given by the following expression: Gb(x|ξ ) = exp[–b(x + ξ )] a sin[a(x – x)] ⎧⎨ ⎩ sin[a(x – x)] sin[a(x – ξ )], x < x < ξ , sin[a(ξ – x)] sin[a(x – x)], ξ < x < x. http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 6 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 Using this Green’s function, the solution of the boundary value problem can be written as follows: y(x) = ∫ x x Gb(x|ξ )f (ξ ) dξ · 3.2 Laplace transform As we have already said, the Laplace transform converts the differential equation with constant coefficients into an algebraic equation whose solution is F(s) = sy() + y′() (s + b) + a – G(s) (s + b) + a , where F(s) and G(s) are the corresponding Laplace transforms of y(x) and f (x), respec- tively. Using the inverse Laplace transform and the convolution product, we get y(x) = y()e–bx cos ax a + y′()e–bx sin ax a –  a ∫ x  e–b(x–ξ ) sin [ a(x – ξ ) ] f (ξ ) dξ . Thus, substituting the boundary conditions, we obtain two algebraic equations, a system involving y() and y′(). Solving this algebraic system, we obtain y(x) = –/a sin[a(x – x)] ∫ x x e–b(x+t) sin [ a(x – x) ] sin [ a(x – t) ] f (t) dt – /a sin[a(x – x)] ∫ x x e–b(x+t) sin [ a(x – t) ] sin [ a(x – x) ] f (t) dt which can be rewritten as follows: y(x) = ∫ x x Gb(x|t)f (t) dt, where the Green’s function is given by Gb(t|x) = exp[–b(x + t)] a sin[a(x – x)] ⎧⎨ ⎩ sin[a(x – x)] sin[a(t – x)], x < t < x, sin[a(x – t)] sin[a(x – x)], x < t < x, which is the same expression as that obtained in Section .. At this point we conclude that for a problem involving initial conditions, the Laplace integral transform is more convenient since y() and y′() are known. On the other hand, i.e., for a problem involving boundary conditions, the Sturm-Liouville, as opposed to the Laplace integral transform, is more convenient in the sense that the calculation is much more simple. http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 7 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 4 Fractional Green’s function Fractional calculus is one of the most accurate tools to refine the description of natural phenomena. The usual way to use this tool is to replace the integer-order derivatives of the partial differential equation that describes one specific phenomenon by a derivative of non-integer order. For many expected reasons, the solution of a fractional partial differ- ential equation used to be much more complicated than the solution of the corresponding integer-order partial differential equation. On the other hand, many important results and generalizations were obtained using this procedure in several areas such as fluid flow, diffuse transport, electrical networks, prob- ability, biomathematics and others [–]. Here, as a generalization to the integer case, we present a calculation associated with the so-called fractional one-sided and two-sided Green’s function relative to the fractional differential equation with constant coefficients, i.e., we obtain the fractional Green’s function for the fractional differential equation whose coefficients are constants. We discuss the problem by means of the Laplace integral trans- form, and as an application, we present explicitly the Green’s function associated with the fractional harmonic oscillator. 4.1 Fractional one-sided Green’s function Let a, b and c be real constants. We present the solution of the fractional differential equa- tion aDα x y(x) + bDβ x y(x) + cy(x) = f (x), () where  < α ≤  and  < β ≤  and the fractional derivatives are taken in the Caputo sense []. We also consider y() =  = y′() as the initial conditions. In the case where α =  and β = , we recover the results associated with the integer case, and taking a =  and b �= , we recover the equation associated with the fractional relaxor-oscillator as dis- cussed in []. Introducing the Laplace transform and using the initial conditions, we obtain an alge- braic equation whose solution can be written as follows: L [ y(x) ] = F(s) asα + bsβ + c , where F(s) is the Laplace transform of the f (x). This expression can be manipulated, using the geometric series, to obtain L [ y(x) ] = F(s) a ∞∑ k= ( – b a )k sβk (sα + c/a)k+ which is valid for |s–λc/a| < . Using the Laplace transform of the generalized Mittag-Leffler function and its corre- sponding inverse [], L [ tμ–Eρ ν,μ ( λtν )] = sνρ–μ (sν – λ)ρ http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 8 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 with Re(s) > , Re(μ) > , λ ∈C and |λs–ν | < , we have y(x) =  a ∞∑ k= ( – b a )k ∫ x  f (ξ )(x – ξ )(α–β)k+α–Ek+ α,(α–β)k+α [ – c a (x – ξ )α ] dξ which is the solution of the fractional differential equation. Thus, the one-sided fractional Green’s function can be written as follows: G(x|ξ ) =  a (x – ξ )α–Eα,α [ – c a (x – ξ )α ] +  a ∞∑ k= ( – b a )k (x – ξ )(α–β)k+α–Ek+ α,(α–β)k+α [ – c a (x – ξ )α ] . () Taking b = , a =  and c = ω in Eq. (), we get G(x|ξ ) = (x – ξ )α–Eα,α [ –ω(x – ξ )α ] which is the fractional one-sided Green’s function associated with the fractional harmonic oscillator. The one-sided Green’s function associated with the classical harmonic oscillator is recovered by introducing α =  in the last equation, i.e., G(x|ξ ) = (x – ξ )E, [ –ω(x – ξ )], x > ξ , which can also be written as follows: G(x|ξ ) =  ω sin [ ω(x – ξ ) ] , which is the same expression as that obtained in Eq. () in the case b = . To conclude this section, we plot, in Figure , a graphic t = x – ξ × tα–Eα,α(–tα) for particular values of the parameter α to compare with the sine function. This graphic is plotted using the program (MatLab Ra). For reader interested in the Mittag-Leffler Figure 1 Graphics for f (t) = tα–1Eα,α (–tα ) × t with t ≡ x – ξ > 0. http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 9 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 function, we suggest a recent nice book [] and the paper [] particularly regarding the asymptotic algebraic behavior of this function. 4.2 Fractional two-sided Green’s function Let a, b and c be real constants. We present the solution of the fractional differential equa- tion, Eq. (), with the homogeneous boundary conditions, y(x) =  = y(x). By means of the Laplace transform, we can write L [ y(x) ] = F(s) + sα–y() + sα–y′() asα + bsβ + c , where F(s) is the Laplace transform of the f (x). This algebraic equation can be manipulated as follows: L [ y(x) ] = F(s) a ∞∑ k= ( – b a )k sβk (sα + c/a)k+ + y() a ∞∑ k= ( – b a )k sβk+α– (sα + c/a)k+ + y′() a ∞∑ k= ( – b a )k sβk+α– (sα + c/a)k+ · From the inverse Laplace transform and the convolution theorem, we get y(x) =  a ∞∑ k= ( – b a )k ∫ x  f (x – ξ )ξ (α–β)k+α–Ek+ α,(α–β)k+α ( – c a ξα ) dξ + y() a ∞∑ k= ( – b a )k L – [ sβk+α– (sα + c/a)k+ ] + y′() a ∞∑ k= ( – b a )k L – [ sβk+α– (sα + c/a)k+ ] which can be rewritten in the following way: y(x) =  a ∞∑ k= ( – b a )k ∫ x  f (ξ )(x – ξ )(α–β)k+α–Ek+ α,(α–β)k+α [ – c a (x – ξ )α ] dξ + y() a ∞∑ k= ( – b a )k x(α–β)kEk+ α,(α–β)k+ ( – c a xα ) + y′() a ∞∑ k= ( – b a )k x(α–β)k+Ek+ α,(α–β)k+ ( – c a xα ) . As we have already said in Section , to explicitly calculate the solution, we must substi- tute the homogeneous boundary conditions in the last equation and determine the y() and y′(). Finally, we can get the respective fractional two-sided Green’s function. We have shown this is a hard calculation. To obtain the results associated with the fractional har- monic oscillator, we introduce b = , a =  and c = ω in the last equation. Remember that to obtain the respective Green’s function, we substitute f (x) to the corresponding delta function. 5 Concluding remarks We have presented and discussed the so-called one-sided and two-sided Green’s function to study, respectively, an initial value problem and a boundary value problem. Besides that, http://www.boundaryvalueproblems.com/content/2013/1/45 Camargo et al. Boundary Value Problems 2013, 2013:45 Page 10 of 11 http://www.boundaryvalueproblems.com/content/2013/1/45 we studied the same problems by means of the Laplace transform methodology in order to conclude which methodology was most accurate for each problem. We also obtained the fractional generalization of the one-sided and two-sided Green’s function in terms of the generalized Mittag-Leffler function. We conclude that for the initial value problem, the Laplace integral transform methodol- ogy is more convenient; on the other hand, for the boundary value problem, the two-sided Green’s function provides a much more simple calculation. It is important to note that in the present manuscript we did not consider the problem involving physical dimensions. This problem has been discussed in a recent paper by Inizan []. A natural continuation of this work would be to study the problems involving partial differential equations with non-constant coefficients and their fractional versions, which could provide a better description of the phenomena related to those equations. Study in this direction is upcoming. Competing interests The authors declare that they have no competing interests. Authors’ contributions RFC had the idea of the work. ECO had the idea to consider the fractional version. AOC had the fractional harmonic oscillator, idea. All authors have made the calculations involved. All authors read and approved the final manuscript. Author details 1Departamento de Matemática, Faculdade de Ciências, Unesp, Bauru, SP 17033-369, Brazil. 2Departamento de Matemática, Imecc, Unicamp, Campinas, SP 13081-970, Brazil. Acknowledgements We are grateful to Prof. J. Vaz Jr., Dr. J. Emílio Maiorino and Dr. E. Contharteze Grigoletto for several useful discussions. Besides that, we are thankful to the referees for several important suggestions which improved this article a lot. Endnote a A recent historical review about George Green can be found in ref. [17]. Received: 16 October 2012 Accepted: 14 February 2013 Published: 6 March 2013 References 1. Capelas de Oliveira, E: Special Functions and Applications, 2nd edn. Editora Livraria da Física, São Paulo (2012) (in Portughese) 2. 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Boundary Value Problems 2013 2013:45. http://www.boundaryvalueproblems.com/content/2013/1/45 One-sided and two-sided Green's functions Abstract Introduction One-sided Green's function Constant coefficients Laplace transform Two-sided Green's function Constant coefficients Laplace transform Fractional Green's function Fractional one-sided Green's function Fractional two-sided Green's function Concluding remarks Competing interests Authors' contributions Author details Acknowledgements Endnote References << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /PageByPage /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /LeaveColorUnchanged /DoThumbnails true /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams true /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments false /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV /HUN /ITA /JPN /KOR /LTH /LVI /NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. 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