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Fortran ii computer program for evaluation of donnell type differential equation for orthotropic circular-conical shell
Difference methods for asymptotic estimates of errors at numerical integration of systems of ordinary differential equations
We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Bibliographical survey of research on stability theory of solutions to partial differential equations
Solution of second order differential equation of hyperbolic type by reduction to system of integral equations
Literature review on stability of solutions to partial differential equations
Boundary conditions for Tricomi equation in symmetric positive form, showing analytical and finite difference solutions
The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.
Lie series solutions for 33 partial differential equations resulting from separation of Helmholtz equation into 11 coordinate systems
Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.
Approximate solution to nonlinear differential equation of third order with bogolyubov asymptotic method
Asymptotic fixed point theorems and periodic systems of functional-differential equations
Ascent existence and uniqueness proof for linear ordinary differential equations
Representation formula for solutions of second order linear differential equation
Existance and stability of solutions to nonlinear operator differential equation
Existence theorems for solutions to second order ordinary differential equations
Boundary value problem for elliptic-parabolic partial differential equations in theory of random processes, obtaining analytical solution in terms of hypergeometric functions
Uniqueness theorem and successive approximations for delay functional differential equations, noting scalar problems