In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations.
A first-order linear system of ODEs is a system in which every equation is first order and depends on the unknown functions linearly. Here we consider systems with an equal number of unknown functions and equations. These may be written as
where is a positive integer, and are arbitrary functions of the independent variable t. A first-order linear system of ODEs may be written in matrix form:
or simply
.
A linear system is said to be homogeneous if for each and for all values of , otherwise it is referred to as non-homogeneous. Homogeneous systems have the property that if are linearly independent solutions to the system, then any linear combination of these, , is also a solution to the linear system where are constant.
The case where the coefficients are all constant has a general solution: , where is an eigenvalue of the matrix with corresponding eigenvectors for . This general solution only applies in cases where has n distinct eigenvalues, cases with fewer distinct eigenvalues must be treated differently.
For an arbitrary system of ODEs, a set of solutions are said to be linearly-independent if:
is satisfied only for .
A second-order differential equation may be converted into a system of first order linear differential equations by defining , which gives us the first-order system:
Just as with any linear system of two equations, two solutions may be called linearly-independent if implies , or equivalently that is non-zero. This notion is extended to second-order systems, and any two solutions to a second-order ODE are called linearly-independent if they are linearly-independent in this sense.
Like any system of equations, a system of linear differential equations is said to be overdetermined if there are more equations than the unknowns. For an overdetermined system to have a solution, it needs to satisfy the compatibility conditions. [1] For example, consider the system:
Then the necessary conditions for the system to have a solution are:
See also: Cauchy problem and Ehrenpreis's fundamental principle.
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Perhaps the most famous example of a nonlinear system of differential equations is the Navier–Stokes equations. Unlike the linear case, the existence of a solution of a nonlinear system is a difficult problem (cf. Navier–Stokes existence and smoothness.)
Other examples of nonlinear systems of differential equations include the Lotka–Volterra equations.
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A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields.
For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., for a form to be exact, it needs to be closed). See integrability conditions for differential systems for more.
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field whose value at a point gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of . If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function may be defined by:
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748, and possibly knew of it as early as 1729.
In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and the determinant are often referred to simply as the Jacobian in literature.
In mathematics, the Hessian matrix, Hessian or Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or, ambiguously, by ∇2.
In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic and parabolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.
The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function. Since a sum of squares must be nonnegative, the algorithm can be viewed as using Newton's method to iteratively approximate zeroes of the components of the sum, and thus minimizing the sum. In this sense, the algorithm is also an effective method for solving overdetermined systems of equations. It has the advantage that second derivatives, which can be challenging to compute, are not required.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.
In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: . It is often important to know these vectors in linear algebra. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor .
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.
In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. They are named after the Soviet mathematician Boris Galerkin.
In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after Carl Gustav Jacob Jacobi.
In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics.
Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form
In classical mechanics, holonomic constraints are relations between the position variables that can be expressed in the following form:
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others.
In control theory, backstepping is a technique developed circa 1990 by Myroslav Sparavalo, Petar V. Kokotovic, and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping.
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable, and, less commonly, in contrast with stochastic differential equations (SDEs) where the progression is random.