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to be prescribed by the user and what is to be left to the computer to calculate to discuss such existence and uniqueness theorems it is necessary to be precise about the domain of the unknown function otherwise speaking only in terms such as a function of two variables it is impossible to meaningfully formulate the results that is the domain of the unknown function must be regarded as part of the structure of the pde itself the following provides two classic examples of such existence and uniqueness theorems even though the two pdes in question are so similar there is a major difference in behavior for the first pde one has the free prescription of a single function while for the second pde one has the free prescription of two functions let b denote the unit radius disk around the origin in the plane for any continuous function u on the unit circle there is exactly one function u on b such that 2 u x 2 2 u y 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 0 and whose restriction to the unit circle is given by u for any functions f and g on the real line r there is exactly one function u on r 1 1 such that 2 u x 2 2 u y 2 0 displaystyle frac partial 2 u partial x 2 frac partial 2 u partial y 2 0 and with u x 0 f x and u y x 0 g x for all values of x even more phenomena are possible for instance the following pde arising naturally in the field of differential geometry illustrates an example where there is a simple and completely explicit solution formula but with the free choice of only three numbers and not even one function if u is a function on r 2 with x u x 1 u x 2 u y 2 y u y 1 u x 2 u y 2 0 displaystyle frac partial partial x frac frac partial u partial x sqrt 1 left frac partial u partial x right 2 left frac partial u partial y right 2 frac partial partial y frac frac partial u partial y sqrt 1 left frac partial u partial x right 2 left frac partial u partial y right 2 0 then there are numbers a b and c with u x y ax by c in contrast to the earlier examples this pde is nonlinear owing to the square roots and the squares a linear pde is one such that if it is homogeneous the sum of any two solutions is also a solution and any constant multiple of any solution is also a solution definition edit a partial differential equation is an equation that involves an unknown function of n 2 displaystyle n geq 2 variables and some of its partial derivatives 4 that is for the unknown function u u r displaystyle u u rightarrow mathbb r of variables x x 1 x n displaystyle x x_ 1 dots x_ n belonging to the open subset u displaystyle u of r n displaystyle mathbb r n the k t h displaystyle k th order partial differential equation is defined as f d k u d k 1 u d u u x 0 displaystyle f d k u d k 1 u dots du u x 0 where f r n k r n k 1 r n r u r displaystyle f mathbb r n k times mathbb r n k 1 dots times mathbb r n times mathbb r times u rightarrow mathbb r and d displaystyle d is the derivative operator notation edit main article notation for differentiation partial derivatives working in r n displaystyle mathbb r n the partial derivatives of a function u displaystyle u can be denoted by u x i displaystyle frac partial u partial x_ i or with a subscript u x i displaystyle u_ x_ i for multiple derivatives multi index notation can be used thus if α α 1 α n displaystyle alpha alpha _ 1 dots alpha _ n the length of α displaystyle alpha is denoted by α α 1 α n displaystyle alpha alpha _ 1 cdots alpha _ n and the iterated partial is denoted by d α u k u x 1 α 1 x n α n displaystyle d alpha u frac partial k u partial x_ 1 alpha _ 1 cdots partial x_ n alpha _ n in the above definition 5 of a partial differential equation the powers of d displaystyle d are the tensors whose components are the partial derivatives of u displaystyle u for example d k u displaystyle d k u is a tensor having n k displaystyle n k components that that are an arrangement of the set d α u α k displaystyle d alpha u mid alpha k after accounting for commutativity of partial derivatives the greek letter δ denotes the laplace operator if u is a function of n variables then δ u u 11 u 22 u n n displaystyle delta u u_ 11 u_ 22 cdots u_ nn in the physics literature the laplace operator is often denoted by 2 in the mathematics literature 2 u may also denote the hessian matrix of u which is here denoted by d 2 u displaystyle d 2 u classification edit linear and nonlinear equations edit a pde is called linear if it is linear in the unknown and its derivatives for example for a function u of x and y a second order linear pde is of the form a 1 x y u x x a 2 x y u x y a 3 x y u y x a 4 x y u y y a 5 x y u x a 6 x y u y a 7 x y u f x y displaystyle a_ 1 x y u_ xx a_ 2 x y u_ xy a_ 3 x y u_ yx a_ 4 x y u_ yy a_ 5 x y u_ x a_ 6 x y u_ y a_ 7 x y u f x y where a i and f are functions of the independent variables x and y only often the mixed partial derivatives u xy and u yx will be equated but this is not required for the discussion of linearity if the a i are constants independent of x and y then the pde is called linear with constant coefficients if f is zero everywhere then the linear pde is homogeneous otherwise it is inhomogeneous this is separate from asymptotic homogenization which studies the effects of high frequency oscillations in the coefficients upon solutions to pdes nearest to linear pdes are semi linear pdes where only the highest order derivatives appear as linear terms with coefficients that are functions of the independent variables the lower order derivatives and the unknown function may appear arbitrarily for example a general second order semi linear pde in two variables is a 1 x y u x x a 2 x y u x y a 3 x y u y x a 4 x y u y y f u x u y u x y 0 displaystyle a_ 1 x y u_ xx a_ 2 x y u_ xy a_ 3 x y u_ yx a_ 4 x y u_ yy f u_ x u_ y u x y 0 in a quasilinear pde the highest order derivatives likewise appear only as linear terms but with coefficients possibly functions of the unknown and lower order derivatives a 1 u x u y u x y u x x a 2 u x u y u x y u x y a 3 u x u y u x y u y x a 4 u x u y u x y u y y f u x u y u x y 0 displaystyle a_ 1 u_ x u_ y u x y u_ xx a_ 2 u_ x u_ y u x y u_ xy a_ 3 u_ x u_ y u x y u_ yx a_ 4 u_ x u_ y u x y u_ yy f u_ x u_ y u x y 0 many of the fundamental pdes in physics are quasilinear such as the einstein equations of general relativity and the navier stokes equations describing fluid motion a pde without any linearity properties is called fully nonlinear and possesses nonlinearities on one or more of the highest order derivatives an example is the monge ampère equation which arises in differential geometry 6 second order equations edit the elliptic parabolic hyperbolic classification provides a guide to appropriate initial and boundary conditions and to the smoothness of the solutions assuming u xy u yx the general linear second order pde in two independent variables has the form a u x x 2 b u x y c u y y lower order terms 0 displaystyle au_ xx 2bu_ xy cu_ yy cdots mbox lower order terms 0 where the coefficients a b c may depend upon x and y if a 2 b 2 c 2 0 over a region of the xy plane the pde is second order in that region this form is analogous to the equation for a conic section a x 2 2 b x y c y 2 0 displaystyle ax 2 2bxy cy 2 cdots 0 more precisely replacing x by x and likewise for other variables formally this is done by a fourier transform converts a constant coefficient pde into a polynomial of the same degree with the terms of the highest degree a homogeneous polynomial here a quadratic form being most significant for the classification just as one classifies conic sections and quadratic forms into parabolic hyperbolic and elliptic based on the discriminant b 2 4 ac the same can be done for a second order pde at a given point however the discriminant in a pde is given by b 2 ac due to the convention of the xy term being 2 b rather than b formally the discriminant of the associated quadratic form is 2 b 2 4 ac 4 b 2 ac with the factor of 4 dropped for simplicity b 2 ac 0 elliptic partial differential equation solutions of elliptic pdes are as smooth as the coefficients allow within the interior of the region where the equation and solutions are defined for example solutions of laplace s equation are analytic within the domain where they are defined but solutions may assume boundary values that are not smooth the motion of a fluid at subsonic speeds can be approximated with elliptic pdes and the euler tricomi equation is elliptic where x 0 by a change of variables the equation can always be expressed in the form u x x u y y 0 displaystyle u_ xx u_ yy cdots 0 where x and y correspond to changed variables this justifies laplace equation as an example of this type 7 b 2 ac 0 parabolic partial differential equation equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables solutions smooth out as the transformed time variable increases the euler tricomi equation has parabolic type on the line where x 0 by change of variables the equation can always be expressed in the form u x x 0 displaystyle u_ xx cdots 0 where x correspond to changed variables this justifies the heat equation which is of the form u t u x x 0 textstyle u_ t u_ xx cdots 0 as an example of this type 7 b 2 ac 0 hyperbolic partial differential equation hyperbolic equations retain any discontinuities of functions or derivatives in the initial data an example is the wave equation the motion of a fluid at supersonic speeds can be approximated with hyperbolic pdes and the euler tricomi equation is hyperbolic where x 0 by change of variables the equation can always be expressed in the form u x x u y y 0 displaystyle u_ xx u_ yy cdots 0 where x and y correspond to changed variables this justifies the wave equation as an example of this type 7 if there are n independent variables x 1 x 2 x n a general linear partial differential equation of second order has the form l u i 1 n j 1 n a i j 2 u x i x j lower order terms 0 displaystyle lu sum _ i 1 n sum _ j 1 n a_ i j frac partial 2 u partial x_ i partial x_ j quad text lower order terms 0 the classification depends upon the signature of the eigenvalues of the coefficient matrix a i j elliptic the eigenvalues are all positive or all negative parabolic the eigenvalues are all positive or all negative except one that is zero hyperbolic there is only one negative eigenvalue and all the rest are positive or there is only one positive eigenvalue and all the rest are negative ultrahyperbolic there is more than one positive eigenvalue and more than one negative eigenvalue and there are no zero eigenvalues 8 the theory of elliptic parabolic and hyperbolic equations have been studied for centuries largely centered around or based upon the standard examples of the laplace equation the heat equation and the wave equation however the classification only depends on linearity of the second order terms and is therefore applicable to semi and quasilinear pdes as well the basic types also extend to hybrids such as the euler tricomi equation varying from elliptic to hyperbolic for different regions of the domain as well as higher order pdes but such knowledge is more specialized systems of first order equations and characteristic surfaces edit see also first order partial differential equation the classification of partial differential equations can be extended to systems of first order equations where the unknown u is now a vector with m components and the coefficient matrices a ν are m by m matrices for ν 1 2 n the partial differential equation takes the form l u ν 1 n a ν u x ν b 0 displaystyle lu sum _ nu 1 n a_ nu frac partial u partial x_ nu b 0 where the coefficient matrices a ν and the vector b may depend upon x and u if a hypersurface s is given in the implicit form φ x 1 x 2 x n 0 displaystyle varphi x_ 1 x_ 2 ldots x_ n 0 where φ has a non zero gradient then s is a characteristic surface for the operator l at a given point if the characteristic form vanishes q φ x 1 φ x n det ν 1 n a ν φ x ν 0 displaystyle q left frac partial varphi partial x_ 1 ldots frac partial varphi partial x_ n right det left sum _ nu 1 n a_ nu frac partial varphi partial x_ nu right 0 the geometric interpretation of this condition is as follows if data for u are prescribed on the surface s then it may be possible to determine the normal derivative of u on s from the differential equation if the data on s and the differential equation determine the normal derivative of u on s then s is non characteristic if the data on s and the differential equation do not determine the normal derivative of u on s then the surface is characteristic and the differential equation restricts the data on s the differential equation is internal to s a first order system lu 0 is elliptic if no surface is characteristic for l the values of u on s and the differential equation always determine the normal derivative of u on s a first order system is hyperbolic at a point if there is a spacelike surface s with normal ξ at that point this means that given any non trivial vector η orthogonal to ξ and a scalar multiplier λ the equation q λξ η 0 has m real roots λ 1 λ 2 λ m the system is strictly hyperbolic if these roots are always distinct the geometrical interpretation of this condition is as follows the characteristic form q ζ 0 defines a cone the normal cone with homogeneous coordinates ζ in the hyperbolic case this cone has nm sheets and the axis ζ λξ runs inside these sheets it does not intersect any of them but when displaced from the origin by η this axis intersects every sheet in the elliptic case the normal cone has no real sheets analytical solutions edit separation of variables edit main article separable partial differential equation linear pdes can be reduced to systems of ordinary differential equations by the important technique of separation of variables this technique rests on a feature of solutions to differential equations if one can find any solution that solves the equation and satisfies the boundary conditions then it is the solution this also applies to odes we assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter and then see if this can be made to solve the problem 9 in the method of separation of variables one reduces a pde to a pde in fewer variables which is an ordinary differential equation if in one variable these are in turn easier to solve this is possible for simple pdes which are called separable partial differential equations and the domain is generally a rectangle a product of intervals separable pdes correspond to diagonal matrices thinking of the value for fixed x as a coordinate each coordinate can be understood separately this generalizes to the method of characteristics and is also used in integral transforms method of ch...
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