pde notes 3

Professor H. M. Atassi CLASS NOTES ON QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS 1. INTRODUCTION De¯nition 1: An equation containing partial derivatives of the unknown function u is said to be an n-th order equation if it contains at least one n-th order derivative, but contains no derivative of order higher than n. De¯nition 2: A partial di®erential equation is said to be linear if it is linear with respect to the unknown function and its derivatives that appear in it. De¯nition 3: A partial di®erential equation is said to be quasilinear if it is linear with respect to all the highest order derivatives of the unknown function. Example 1: The equation @2u @x2+a(x; y)@2u @y2¡2u= 0 is a second order linear partial di®erential equation. However, the following equation @u @x@2u @x2+@u @y@2u @y2+u2= 0 is a second order quasilinear partial di®erential equation. Finally, the equation (@u @x)2+ (@u @y)2¡u= 0 is a ¯rst order partial di®erential equation which is neither linear nor quasilinear. De¯nition 4: A solution of a partial di®erential equation is any function that, when substituted for the unknown function in the equation, reduces the equation to an identity in the unknown variables. Example 2: Let us consider the one dimensional wave equation (@2 @t2¡c2@2 @x2)u= 0: It is well known, and will be shown later, that the general solution of this equation can be cast as u=f(x¡ct) +g(x+ct); where f and g are arbitrary twice di®erentiable functions of the single variables »= x¡ctand´=x+ct, respectively. It is easy to see, using the chain rule, that @2u @t2=c2(d2f dx2+d2g dy2); @2u @x2=d2f dx2+d2g dy2: 1 Substitution into the wave equation leads to an identity. De¯nition 5: LetÁ(~ x) be a function of the vector ~ x= (x1; x2; : : : ; x n) having ¯rst order derivatives. Then the vector in Rn (D1; D2; : : : ; D n)Á; where Di=@ @xi;is called the gradient of Áand usually denoted as0grad Á0or0rÁ0. De¯nition 6: Let~ vbe a unit vector in Rnandvmeasures the distance on ~ v. Then the limit dÁ(~ x) dv= lim ¢t!0Á(~ x+~ v¢t)¡Á(~ x) ¢t; if it exists, is called the derivative of Áin the ~ vdirection. It is easy to show that dÁ(~ x) dv=rÁ(~ x)¢~ v= (v1D1; v2D2; : : : ; v nDn)Á(~ x); where ~ v= (v1; v2; : : : ; v n). Example 3: (i) In R2,~ x= (x; y). Then grad Á (x; y) = (Dx; Dy)Á= (@Á @x;@Á @y): The derivative of Áin the direction ~ v= (1=p 2;1=p 2), the ¯rst bisector of the plane x-y, is dÁ dv=1p 2(@Á @x+@Á @y): (ii) Let Á(x; y; z ) be a function having continuous ¯rst order derivatives. The equation Á(x; y; z ) =c; where c2R, represents a surface S. As we move along a path °(s) on the surface S, dÁ ds=@Á @xdx ds+@Á @ydy ds+@Á @zdz ds= 0; dÁ ds=grad Á ¢(dx ds;dy ds;dz ds) = 0 : For an in¯nitesimal change in ds, the vector ( dx=ds; dy=ds; dz=ds ) is in the plane tangent to the surface, at the point M(x; y; z ). Consequently, grad Á is orthogonal to the surface Á=c. 2 2. FIRST ORDER QUASILINEAR PARTIAL DIFFERENTIAL EQUA- TIONS We restrict our exposition to ¯rst order quasilinear partial di®erential equations (FO- QPDE) with two variables, since this case a®ords a real geometric interpretation. However, the treatment can be extended without di±culty to higher order spaces. The general form of FOQPDE with two independent variables is a(x; y; u )@u @x+b(x; y; u )@u @y=c(x; y; u ) (2.1) where a,b, and care continuous functions with respect to the three variables x,y,u. Letu=u(x; y) be a solution to equation (2 :1). If we identify uwith the third coordinate zinR3, then u=u(x; y) represents a surface S. The direction of the normal to S is the vector ( ux; uy;¡1), where ux,uyare short hand notation of@u @x and@u @y, respectively. On the other hand, equation (2 :1) can be written as the inner product (ux; uy;¡1)¢(a; b; c ) = 0 : (2.2) Thus, ( a; b; c ) is perpendicular to the normal to S and consequently, must lie in the plane tangent to S. Let us consider a path °(s) on S. The rate of variation of uas we move along °is du ds=@u @xdx ds+@u @ydy ds: (2.3) The vector ( dx=ds; dy=ds; du=ds ) is naturally the tangent to the curve °(s). Equation (2:3) can be rewritten as the inner product (ux; uy;¡1)¢(dx=ds; dy=ds; du=ds ) = 0 : (2.4) Comparing (2 :2) and (2 :4), we see that there is a particular family of curves on the surface S, de¯ned by dx=ds a=dy=ds b=du=ds c: (2.5) These curves are called characteristics and will be denoted by C(s), or simply C. We note that there are actually only two independent equations in the system (2 :5), therefore, its solutions comprise in all a two-parameter family of curves in space. Theorem 1: Any one parameter subset of the characteristics generates a solution of the ¯rst order quasilinear partial di®erential equation (2 :1). Proof: Let u=u(x; y) be the surface generated by a one-parameter family C(s) of the characteristics whose di®erential equations are (2 :5). By taking the rate of variation ofualong a characteristic curve C(s), we get du ds=uxdx ds+uydy ds; or (ux; uy;¡1)¢(dx=ds; dy=ds; du=ds ) = 0 : (2.6) 3 But equations (2 :5) state that the vectors ( dx=ds; dy=ds; du=ds ) and ( a; b; c ) are collinear. Therefore, (ux; uy;¡1)¢(a; b; c ) = 0 ; or aux+buy=c; and we are done. Corollary 1: The general solution to equation (2 :1) is de¯ned by a single relation between two arbitrary constants occurring in the general solution of the system of ordinary di®erential equations (dx=ds ) a=(dy=ds ) b=(du=ds ) c; or, in other words, by any arbitrary function of one independent variable. Example 1: Consider the ¯rst order partial di®erential equation, @u @t+u@u @x= 0: The characteristics are de¯ned by (dt=ds ) 1=(dx=ds ) u=(du=ds ) 0: The last equation gives immediately u=k1. Since uis constant along a given characteristic, then the ¯rst equation can be integrated immediately: x¡k1t=k2 The general solution is then k1=f(k2); where fis an arbitrary function of the single variable k2. Substituting k1andk2by their expressions, we ¯nally have u=f(x¡ut): Example 2: Consider the equation 3@u @x¡7@u @y= 0: The characteristics are solution of the system (dx=ds ) 3=(dy=ds ) ¡7=(du=ds ) 0: 4 By integration, we get u=k1; 3y+ 7x=k2: The characteristics are straight lines intersection of the two families of planes de¯ned by these equations. Any arbitrary relation between k1andk2is a solution. The general solution is then u=f(3y+ 7x); where fis an arbitrary function of the one independent variable 3 y+ 7x. Example 3: Consider the equation y@u @x¡x@u @y= 0: The characteristics are solution of the system (dx=ds ) y=(dy=ds ) ¡x=(du=ds ) 0: By integration, we get u=k1; x2+y2=k2: The characteristics are circles located in the plane u=k1. The general solution is u=f(x2+y2); where fis an arbitrary function of the independent variable ( x2+y2). Geometrically, the general solution is any surface of revolution around the u-axis. Remark 1: The previous examples illustrate the application of the theory of char- acteristics to ¯nd the general solution to a ¯rst order quasilinear partial di®erential equation. Given the simple forms of these examples, the student could become sus- picious that in the general case it will not be possible to get a closed analytical form of the solution. A careful examination of system (2 :5) su±ces to convince you that, in the general case, i.e., when a,b, and care arbitrary functions of x,y, and u, the system of ordinary di®erential equations (2 :5) cannot be integrated analytically. Nevertheless, for a given initial or boundary value problem, system (2 :5) provides a numerical solution. Remark 2: When equation (2 :1) is a linear homogeneous partial di®erential equation a(x; y)ux+b(x; y)uy=c(x; y)u: (2.7) The solutions form a vector space which we call the solution space. This solution space is naturally the null space of the linear operator a(x; y)Dx+b(x; y)Dy¡c(x; y): (2.8) 5 It should be pointed out that since the general solution to (2 :7) is any arbitrary func- tion of one independent variable, there is a nondenumerable in¯nity of independent solutions to equations (2 :7). Therefore, the dimension of null space of the operator (2:8) is in¯nity. This result appears to be in sharp contrast with what we know about the ¯rst order ordinary linear di®erential operators where the null space is one- dimensional. One may also add that this augurs the di±culties we shall encounter in the study of partial di®erential operators. THE BOUNDARY VALUE PROBLEM FOR A FIRST ORDER PARTIAL DIFFERENTIAL EQUATION The theory of characteristics enables us to de¯ne the solution to FOQPDE (2 :1) as surfaces generated by the characteristic curves de¯ned by the ordinary di®erential equations (2 :5). However, a physical problem is not uniquely speci¯ed if we simply give the di®erential equation which the solution must satisfy, for, as we have seen, there are an in¯nite number of solutions of every equation. In order to make the problem a de¯nite one, with a unique answer, we must pick out of the mass of pos- sible solutions, the one which has certain de¯nite properties along de¯nite boundary surfaces. These properties represent the boundary conditions which the solution must satisfy. The ¯rst fact which we must notice is that we cannot try to make the solutions of a given equation satisfy any sort of boundary conditions; for there is a de¯nite set of boundary conditions which will give nonunique or impossible answers. The study of the proper boundary conditions to be speci¯ed on de¯nite boundary curves or sur- faces is often termed the Cauchy problem, in honor of the French mathematician who laid the foundations to our present knowledge in this area. Statement of the Cauchy problem for FOQPDE Let°(t) be a curve in region R of the x-yplane, where the value of the function u(x; y) which satis¯es equation (2 :1) is speci¯ed as u(t). What are the conditions to be satis¯ed by °(t) and u(t), in order that the boundary value problem so de¯ned has a unique solution? Consider an initial curve °(t) in a region R »=»(t); (2.9) ´=´(t): Since u(x; y) is speci¯ed on °(t) as u(x; y) =u(t) (2.10) then the three functions »=»(t); ´=´(t); u=u(t) (2.11) 6 de¯ne a curve ¡ in space whose projection over the x-yplane is °. By a proper choice of the parameter s, the equations of the characteristics are dx ds=a;dy ds=b;du ds=c; (2.12) The characteristic curve passing by a point M(t)2¡ is then de¯ned by the following equations x=x(s; t); y=y(s; t); u=u(s; t): (2.13) Initial Cur ve Γ(t)C(s) Charac teristics Figure 2.1: Solution of the boundary-value problem of a ¯rst-order partial di®erential equation by a one-parameter family of characteristics. Equation (2 :13) represent the surface S in a parametric form. When tis kept constant we move along a characteristic curve C(s). For a given value of s, for example s0, we move on the surface S along the curve ¡. Let us now assume that the initial curve ¡ was a characteristic curve C. It is then obvious that equations (2 :9) satisfy system (2 :12) and there will be no surface solution generated this way. We conclude that in order to generate a surface solution, ¡ should not be a characteristic curve. Mathematically, this means that the two vectors (d»=dt; d´=dt; du=dt ) and ( a; b; c ) must be linearly independent. A su±cient condition to insure this linearly independence is that the determinant J=¯¯¯¯¯¯¯d» dtd´ dt a b¯¯¯¯¯¯¯(2.14) never vanishes on ¡. An alternative form to the condition (2 :14) can be obtained from equations (2 :13). The vectors tangent to C and ¡ are respectively ( xs; ys; us) and ( xt; yt; ut). A su±cient condition for their independence is that the Jacobian J=¯¯¯¯¯¯¯xsys xtyt¯¯¯¯¯¯¯ 7 never vanishes on ¡. We are then led to the following theorem: Theorem 2: The solution u(x; y) may be freely speci¯ed along a curve °in R and the resulting speci¯cation determines a unique solution of (2 :1) if and only if °intersects the projection on the x-yplane of each characteristic curve exactly once. Example 4: Take the simple equation ux+uy=c(x; y; u ) (2.15) and take Ras the square region in Figure 2.2. The projection on the x-y plane of the characteristic curves (often called the characteristics) are the lines x¡y=¸, for x; y2R. A few of these characteristics are shown in Figure 2.2 (a). We see that the curve °1of Figure 2.2 (b) meets the conditions of theorem 2 so that speci¯cation of u(x; y) along °1converts (2.1) into a boundary-value problem with a unique solution. However, Curve °2of Figure 2.2(c) while not intersecting any characteristic more than once does not intersect all the characteristics in R, so the speci¯cation of u(x; y) along °2would specify u(x; y) =u(t) only for those characteristics which intersect °2, i.e., inside a sub region of Rbetween the limiting characteristics shown in Figure 2.2(c), which is called domain of in°uence of the curve °2. Speci¯cation of u(x; y) along °2, then determines v only in the domain of in°uence of °2and we would say such a boundary-value problem was understated for the whole region R. The curve °3of Figure 2.2(d)intersects some characteristics more than once, so speci¯cation of u(x; y) along v would convert (2.1) to a boundary-value problem with no solution unless we were lucky enough that the speci¯ed values for the several points on the same characteristics were compatible with the general solution; we say, this boundary- value problem is overspeci¯ed . (a)(b) (c) (d) 1 1 1 11 1 1 1 γ1γ2γ3 Figure 2.2: (a) Characteristics of of (2.15), (b) Well-posed boundary-value problem, (c) Understated boundary-value problem, (d) Overspeci¯ed boundary-value problem. Theorem 3: The boundary-value problem for FOQPDE aux+buy=c; x; y; in R 8 u(x; y) =u(t); x; y; on ° has a unique solution for any speci¯ed u(t) if and only if °intersects the projection of each characteristic curve in R exactly once. Remark 3: Due to the nonlinearity of (2 :1), the domain of in°uence has only local signi¯cance. The geometry of the characteristics can become quite complicated to the point that a well posed boundary-value problem could be very di±cult to solve as we move along the characteristics. 3. SYSTEMS OF FIRST ORDER QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS We denote by u,vthe dependent variables (unknown functions) and x,ythe inde- pendent variables. The general form of the system of di®erential equations is L1=A1ux+B1uy+C1vx+D1vy+E1= 0; (3.1) L2=A2ux+B2uy+C2vx+D2vy+E2= 0; in which A1,A2,¢¢¢,E2are known functions of x,y,u,v. We make the assumption that all functions occurring in the theory are continuous and possess as many contin- uous derivatives as may be required. Without restrictions we assume that nowhere A1=A2=B1=B2=C1=C2=D1=D2: IfE1=E2= 0, the system is homogeneous. If the coe±cients A1,A2,¢¢¢,E2are functions of xandyonly, the equations are linear and consequently much easier to handle. De¯nition 1: If the system is homogeneous, E1=E2= 0, and the coe±cients A1, A2,¢¢¢,D2are functions of u,valone, the equations are said to be reducible. The following theorem justi¯es the above de¯nition. Theorem 1: For any region where the Jacobian j=uxvy¡uyvx (3.2) is not zero, a reducible system (3.1) can be transformed by interchanging the roles of the dependent and independent variables into the following equivalent linear system A1yv¡B1xv¡C1yu+D1xu= 0; (3.3) A2yv¡B2xv¡C2yu+D2xu= 0: 9 The transformation of the ( x; y)- plane into the ( u; v)- plane is called a hodograph transformation. It is advantageous, in general, to perform the hodograph transfor- mation whenever the Jacobian j6= 0 in the region where a solution is desired. Characteristic curves and characteristic equations We have shown (De¯nition 1.6) that an expression such as A1ux+B1uyrepresents derivative in a direction dx=dy =A1=B1. Each equation of (3.1) can be thought of as a linear relation between a derivative of uin the direction ( A; B) and a derivative ofvin the direction ( C; D). We now ask for a linear combination L=¸1L1+¸2L2; so that in the di®erential expression L, the derivatives of uand those of vcombine to derivatives in the same direction. Such direction, which depends on the variables x,yas well as u,v, is called characteristic direction. Suppose the direction is given by the ratio x¾=y¾. Then the condition that, in L,u andvare di®erentiated in this direction, is simply ¸1A1+¸2A2 ¸1B1+¸2B2=¸1C1+¸2C2 ¸1D1+¸2D2=x¾ y¾(3.4) The expression Lcan be written after multiplication with either x¾ory¾as (¸1A1+¸2A2)u¾+ (¸1C1+¸2C2)v¾+ (¸1E1+¸2E2)x¾=x¾L (3.5) (¸1B1+¸2B2)u¾+ (¸1D1+¸2D2)v¾+ (¸1E1+¸2E2)y¾=y¾L If at the point ( x,y), the functions uandvsatisfy (3.1), we obtain four homogeneous linear equations for ¸1and¸2: ¸1(A1y¾¡B1x¾) +¸2(A2y¾¡B2x¾) = 0 ; ¸1(C1y¾¡D1x¾) +¸2(C2y¾¡D2x¾) = 0 ; (3.6) ¸1(A1u¾+C1v¾+E1x¾) +¸2(A2u¾+C2v¾+E2x¾) = 0 ; ¸1(B1u¾+D1v¾+E1y¾) +¸2(B2u¾+D2v¾+E2y¾) = 0 : In order to have nontrivial solutions to (3.6), all determinants of two rows in the matrix of the coe±cient of ¸1and¸2must vanish. Thus a number of characteristic relations follow. From the ¯rst two equations, we get ¯¯¯¯¯A1y¾¡B1x¾A2y¾¡B2x¾ C1y¾¡D1x¾C2y¾¡D2x¾¯¯¯¯¯= 0 (3.7) or ~ay2 ¾¡2~bx¾y¾+ ~cx2 ¾= 0; (3.8) 10 where ~a= [AC];2~b= [AD] + [BC];~c= [BD] with the abbreviation [XY] =X1Y2¡X2Y1: Equation (3.8) de¯nes the directions of the characteristics. There are three cases: (i) ~a~c¡~b2>0, then (3.8) cannot be satis¯ed by a real direction. The di®erential equations (3.1) are then called elliptic. (ii) ~a~c¡~b2= 0, then there is only one characteristic direction. The di®erential equations (3.1) are then called parabolic. (iii) ~a~c¡~b2<0, we have two di®erent characteristic directions through each point. The di®erential equations (3.1) are then called hyperbolic. It is obvious that the notion of characteristics is useful only when they do exist. For this reason, we shall from now on assume the hyperbolic character of system (3.1) and accordingly suppose ~a~c¡~b2<0: Let»=y¾=x¾be the slope of the characteristics, then »satis¯es the quadratic equation ~a»2¡2~b»+ ~c= 0: (3.9) This equation has two distinct real solutions »+and»¡. The characteristic curves are the envelopes to these slopes C+for»+andC¡for»¡. Their respective equations are I+:dy d®=»+dx d®along C+; (3.10) I¡:dy d¯=»¡dx d¯along C¡: Let us to return to system (3.6) and write that the determinant of the ¯rst and third equation is nil. ¯¯¯¯¯A1y¾¡B1x¾ A2y¾¡B2x¾ A1u¾+C1v¾+E1x¾A2u¾+C2v¾+E2x¾¯¯¯¯¯= 0: (3.11) After rearrangement, we obtain Tu¾+ (a»¡S)v¾+ (K»¡H)x¾= 0; (3.12) in which, T= [AB]; S= [BC]; K= [AE]; H= [BE]: (3.13) This relation holds on C+if we identify »with »+and¾with ®, and likewise, C¡, if identify »with »¡and¾with ¯. 11 Thus we arrive at the following four characteristic equations: I+y®¡»+x®= 0; I¡y¯¡»¡x¯= 0; II+Tu®+ (~a»+¡S)v®+ (K»+¡H)x®= 0; (3.14) II¡Tu¯+ (~a»¡¡S)v¯+ (K»¡¡H)x¯= 0; It is not overemphasizing to stress again that II+is valid only on C+whose equation isI+andII¡onC¡whose equation is I¡. Remark 1: Unless the system (3.1) is linear, »+and»¡depend on x,yandu,v. Consequently, the characteristics depend on the individual solutions u,v, and so does the hyperbolic character of the system (3.1). Example 1: Let us consider the case of a one-dimensional unsteady isentropic °ow. The unknown functions are the density ½and the velocity u, and the independent variables are x,t. The °ow equations are: ½t+u½x+½ux= 0; (3.15) ut+uux+c2 ½½x= 0: We ¯nd that ~ a= 1, ~b=u, ~c=u2¡c2, ~a~c¡~b2=¡c2<0. Hence there are two characteristic families: x®= (u+c)t®; x ¯= (u¡c)t¯ (3.16) The characteristic equation for ½andubecome u®+c ½½®= 0; u ¯¡c ½½¯= 0: (3.17) Equations (3.16) express the fact that the characteristic curves in the ( x; t)-plane represent motions of possible disturbances called sound waves whose velocities dx dt=u+c;dx dt=u¡c di®er from the particle velocity uby the sound velocity §c. For an isentropic °ow, ( °¡1)d½ ½= 2dc c. This result can be used to eliminate ½from (3.18) and get u®+2 °¡1c®= 0; u ¯¡2 °¡1c¯= 0: (3.18) 12 Equations (3.18) can then be integrated to give u+2 °¡1c=r; u ¡2 °¡1c=s; (3.19) where randsare constant along the characteristics C+andC¡, respectively. rand sare known as the Riemann invariants. If one of the Riemann invariants is constant throughout the domain, the solution corresponds to a wave motion in a one direction onlyand is said to be a simple wave . Proposition: A °ow in a region adjacent to a region of constant state is always a simple wave. Example 2: The equations for steady two-dimensional isentropic irrotational inviscid °ow are ½ux+½uy+u½x+v½y= 0; (3.20) uux+vuy+c2 ½½x= 0; (3.21) uvx+vvy+c2 ½½y= 0: (3.22) The density ½can be eliminated from these equations by multiplying the ¯rst by ¡c2 ½, the second by uand the third by v, and adding them. The resulting equation is (u2¡c2)ux+uvu y+uvvx+ (v2¡c2)vy= 0: (3.23) To this equation we add the condition of irrotational °ow uy¡vx= 0: (3.24) The system of equations (3.20){(3.22) is quasilinear (in fact, it is a reducible system). The equations of the characteristics directions is (c2¡u2)»2+ 2uv»+ (c2¡v2) = 0 : (3.25) The characteristics are real if u2+v2> c2, or if the °uid is supersonic. In this case, the system of characteristic equations can be written as I+:y®=»+x®; I¡:y¯=»¡x¯; II+:u®=¡»+v®; (3.26) II¡:u¯=¡»¡v¯: The characteristic curves determined from I+andI¡are usually called Mach lines. 13 Theorem 2: Consider a smooth curve °(t) where u(t) and v(t) are freely speci¯ed, then the boundary-value problem for the system of di®erential equations L1=A1ux+B1uy+C1vx+D1vy+E1= 0; (3.27) L2=A2ux+B2uy+C2vx+D2vy+E2= 0; in which A1; A2;¢¢¢; E2are known functions of x,y,u,vin the region R bounded by the two characteristics passing though point P and the domain of dependence cut by them from the initial curve °(t) has a unique solution. 3. SECOND ORDER QUASILINEAR PARTIAL DIFFERENTIAL EQUA- TION We denote by © the dependent variable and x,ythe independent variables. The general form of a second order partial di®erential equation is ~a©xx+ 2~b©xy+ ~c©yy+~d= 0; (3.28) in which ~ a,~b, ~c,~dare known functions of x,y, ©, © x, ©y. This problem can be reduced to that of (3.1) by introducing the variables u= © x; v = © y and the fact that for continuously di®erentiable functions uy=vx The equation for the characteristics is ~ay2 ¾¡2~bx¾y¾+ ~cx2 ¾= 0; (3.29) where ~ a,~b, ~care de¯ned as previously. Calculating the relationship between u¾and v¾as in (3.12), we ¯nally arrive at the following characteristic system I+y®¡»+x®= 0; I¡y¯¡»¡x¯= 0; II+u®+»¡v®+x®~d ~a= 0; (3.30) II¡u¯+»+v¯+x¯~d ~a= 0: Canonical form of a second order partial di®erential equation Note that the two families of characteristic I+andI¡can be de¯ned as 14 x=x(®; ¯) (3.31) y=y(®; ¯) (3.32) where ¯is constant along I+and®is constant along I¡. If the Jacobian of the transformation (3.31), (3.32) is not zero, we can de¯ne ®=®(x; y) (3.33) ¯=¯(x; y): (3.34) Using the new variables ®and¯, equation (3.28) becomes ©®;¯=F(®; ¯; ©®;©®;©): (3.35) Equation (3.35) in known as the canonical form of (3.28). 15