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A partial differential equation whose solution does not depend continuously on its parameters (including but not limited to boundary conditions) is said to be ill-posed.

A Fredholm integral equation of the second kind phi(x)=f(x)+lambdaint_a^bK(x,t)phi(t)dt (1) may be solved as follows. Take phi_0(x) = f(x) (2) phi_1(x) = ...

Q_n^((alpha,beta))(x)=2^(-n-1)(x-1)^(-alpha)(x+1)^(-beta) ×int_(-1)^1(1-t)^(n+alpha)(1+t)^(n+beta)(x-t)^(-n-1)dt. In the exceptional case n=0, alpha+beta+1=0, a nonconstant ...

A generalization of the Runge-Kutta method for solution of ordinary differential equations, also called Rosenbrock methods.

The Kuhn-Tucker theorem is a theorem in nonlinear programming which states that if a regularity condition holds and f and the functions h_j are convex, then a solution ...

The second-order ordinary differential equation y^('')+k/xy^'+epsilony^'y=0.

A quantity involving primitive cube roots of unity which can be used to solve the cubic equation.

The scalar form of Laplace's equation is the partial differential equation del ^2psi=0, (1) where del ^2 is the Laplacian. Note that the operator del ^2 is commonly written ...

Let F be a differential field with constant field K. For f in F, suppose that the equation g^'=f (i.e., g=intf) has a solution g in G, where G is an elementary extension of F ...

A necessary and sufficient condition for all the eigenvalues of a real n×n matrix A to have negative real parts is that the equation A^(T)V+VA=-I has as a solution where V is ...