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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 characteristics edit main article method of characteristics the characteristic surface in n 2 dimensional space is called a characteristic curve 10 in special cases one can find characteristic curves on which the first order pde reduces to an ode changing coordinates in the domain to straighten these curves allows separation of variables and is called the method of characteristics more generally applying the method to first order pdes in higher dimensions one may find characteristic surfaces integral transform edit an integral transform may transform the pde to a simpler one in particular a separable pde this corresponds to diagonalizing an operator an important example of this is fourier analysis which diagonalizes the heat equation using the eigenbasis of sinusoidal waves if the domain is finite or periodic an infinite sum of solutions such as a fourier series is appropriate but an integral of solutions such as a fourier integral is generally required for infinite domains the solution for a point source for the heat equation given above is an example of the use of a fourier integral change of variables edit often a pde can be reduced to a simpler form with a known solution by a suitable change of variables for example the black scholes equation v t 1 2 σ 2 s 2 2 v s 2 r s v s r v 0 displaystyle frac partial v partial t tfrac 1 2 sigma 2 s 2 frac partial 2 v partial s 2 rs frac partial v partial s rv 0 is reducible to the heat equation u τ 2 u x 2 displaystyle frac partial u partial tau frac partial 2 u partial x 2 by the change of variables 11 v s t v x τ x ln s τ 1 2 σ 2 t t v x τ e α x β τ u x τ displaystyle begin aligned v s t v x tau x ln left s right tau tfrac 1 2 sigma 2 t t v x tau e alpha x beta tau u x tau end aligned fundamental solution edit main article fundamental solution inhomogeneous equations can often be solved for constant coefficient pdes always be solved by finding the fundamental solution the solution for a point source p d u δ displaystyle p d u delta then taking the convolution with the boundary conditions to get the solution this is analogous in signal processing to understanding a filter by its impulse response superposition principle edit further information superposition principle the superposition principle applies to any linear system including linear systems of pdes a common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude for example sin x sin x 2 sin x the same principle can be observed in pdes where the solutions may be real or complex and additive if u 1 and u 2 are solutions of linear pde in some function space r then u c 1 u 1 c 2 u 2 with any constants c 1 and c 2 are also a solution of that pde in the same function space methods for non linear equations edit see also nonlinear partial differential equation there are no generally applicable analytical methods to solve nonlinear pdes still existence and uniqueness results such as the cauchy kowalevski theorem are often possible as are proofs of important qualitative and quantitative properties of solutions getting these results is a major part of analysis nevertheless some techniques can be used for several types of equations the h principle is the most powerful method to solve underdetermined equations the riquier janet theory is an effective method for obtaining information about many analytic overdetermined systems the method of characteristics can be used in some very special cases to solve nonlinear partial differential equations 12 in some cases a pde can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods many interesting problems in science and engineering are solved in this way using computers sometimes high performance supercomputers lie group method edit from 1870 sophus lie s work put the theory of differential equations on a more satisfactory foundation he showed that the integration theories of the older mathematicians can by the introduction of what are now called lie groups be referred to a common source and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration he also emphasized the subject of transformations of contact a general approach to solving pdes uses the symmetry property of differential equations the continuous infinitesimal transformations of solutions to solutions lie theory continuous group theory lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations to find its lax pairs recursion operators bäcklund transform and finally finding exact analytic solutions to the pde symmetry methods have been recognized to study differential equations arising in mathematics physics engineering and many other disciplines semi analytical methods edit the adomian decomposition method 13 the lyapunov artificial small parameter method and his homotopy perturbation method are all special cases of the more general homotopy analysis method 14 these are series expansion methods and except for the lyapunov method are independent of small physical parameters as compared to the well known perturbation theory thus giving these methods greater flexibility and solution generality numerical solutions edit the three most widely used numerical methods to solve pdes are the finite element method fem finite volume methods fvm and finite difference methods fdm as well other kind of methods called meshfree methods which were made to solve problems where the aforementioned methods are limited the fem has a prominent position among these methods and especially its exceptionally efficient higher order version hp fem other hybrid versions of fem and meshfree methods include the generalized finite element method gfem extended finite element method xfem spectral finite element method sfem meshfree finite element method discontinuous galerkin finite element method dgfem element free galerkin method efgm interpolating element free galerkin method iefgm etc finite element method edit main article finite element method the finite element method fem its practical application often known as finite element analysis fea is a numerical technique for approximating solutions of partial differential equations pde as well as of integral equations using a finite set of functions 15 16 the solution approach is based either on eliminating the differential equation completely steady state problems or rendering the pde into an approximating system of ordinary differential equations which are then numerically integrated using standard techniques such as euler s method runge kutta etc finite difference method edit main article finite difference method finite difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives finite volume method edit main article finite volume method similar to the finite difference method or finite element method values are calculated at discrete places on a meshed geometry finite volume refers to the small volume surrounding each node point on a mesh in the finite volume method surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals using the divergence theorem these terms are then evaluated as fluxes at the surfaces of each finite volume because the flux entering a given volume is identical to that leaving the adjacent volume these methods conserve mass by design neural networks edit this section is an excerpt from deep learning partial differential equations edit physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner 17 one example is the reconstructing fluid flow governed by the navier stokes equations using physics informed neural networks does not require the often expensive mesh generation that conventional cfd methods rely on 18 19 it is evident that geometric and physical constraints have a synergistic effect on neural pde surrogates thereby enhancing their efficacy in predicting stable and super long rollouts 20 weak solutions edit main article weak solution weak solutions are functions that satisfy the pde yet in other meanings than regular sense the meaning for this term may differ with context and one of the most commonly used definitions is based on the notion of distributions an example 21 for the definition of a weak solution is as follows consider the boundary value problem given by l u f in u u 0 on u displaystyle begin aligned lu f quad text in u u 0 quad text on partial u end aligned where l u i j j a i j i u i b i i u c u displaystyle lu sum _ i j partial _ j a ij partial _ i u sum _ i b i partial _ i u cu denotes a second order partial differential operator in divergence form we say a u h 0 1 u displaystyle u in h_ 0 1 u is a weak solution if u i j a i j i u j v i b i i u v c u v d x u f v d x displaystyle int _ u bigg sum _ i j a ij partial _ i u partial _ j v sum _ i b i partial _ i u v cuv bigg dx int _ u fvdx for every v h 0 1 u displaystyle v in h_ 0 1 u which can be derived by a formal integral by parts an example for a weak solution is as follows ϕ x 1 4 π 1 x displaystyle phi x frac 1 4 pi frac 1 x is a weak solution satisfying 2 ϕ δ in r 3 displaystyle nabla 2 phi delta text in r 3 in distributional sense as formally r 3 2 ϕ x ψ x d x r 3 ϕ x 2 ψ x d x ψ 0 for ψ c c r 3 displaystyle int _ r 3 nabla 2 phi x psi x dx int _ r 3 phi x nabla 2 psi x dx psi 0 text for psi in c_ c infty r 3 theoretical studies edit in pure mathematics the theoretical studies of pdes focus on the criteria for a solution to exist and the properties of a solution while finding its formula is often secondary well posedness edit main article well posed problem well posedness refers to a common schematic package of information about a pde to say that a pde is well...