# Numerical existence theorem for solutions of two-point boundary value problems of nonlinear differential equations

@article{Takayasu2010NumericalET, title={Numerical existence theorem for solutions of two-point boundary value problems of nonlinear differential equations}, author={Akitoshi Takayasu and Shin'ichi Oishi and Takayuki Kubo}, journal={Nonlinear Theory and Its Applications, IEICE}, year={2010}, volume={1}, pages={105-118} }

In this paper, a numerical method is presented for verifying the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations. Taking into account every error of numerical computations such as the discretization error and the rounding error, this method also provides mathematically guaranteed error bounds between approximations obtained by numerical computations and the exact solution whose existence is proven by the numerical existence…

## 9 Citations

### A Posteriori Error Bounds for Two Point Boundary Value Problems: A Green's Function Approach

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The utility of the computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs) is demonstrated and used to rigorously show the existence of a periodic orbit in the Lorenz system.

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An improved method is presented for the norm estimation of the residual to the operator equation in this refined formulation, piecewise quadratic finite element base functions are used.

### Computable Error Bounds for a Class of Boundary Value Problems: a Coefficients Perturbation Approach

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This paper is concerned with computing upper and lower bounds for the error committed when some boundary value problems are approximated by means of numerical techniques based on the Coefficients…

### Verified solutions of two-point boundary value problems for nonlinear oscillators

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- 2011

: Using techniques introduced by Nakao [4], Oishi [5, 6] and applied by Takayasu, Oishi, Kubo [11, 12] to certain nonlinear two-point boundary value problems (see also Rump [7], Chapter 15), we…

### A numerical verification method for nonlinear functional equations based on infinite-dimensional Newton-like iteration

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We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization…

## References

SHOWING 1-10 OF 16 REFERENCES

### A numerical approach to the proof of existence of solutions for elliptic problems

- Mathematics
- 1988

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the…

### A numerical approach to the proof of existence of solutions for elliptic problems II

- Mathematics
- 1988

This paper is a continuation of the preceding study ([2]) in which we described an automatic proof by computer, utilizing Schauder’s fixed point theorem, of the existence of weak solutions for…

### Computable a Posteriori $L_\infty $-Error Bounds for the Approximate Solution of Two-Point Boundary Value Problems

- Mathematics
- 1975

In this paper we use the general theory of Newton’s method for operator equations with functional constraints in Banach spaces recently developed by Tapia and the Kantorovich theorem to construct…

### Computer-assisted existence proofs for two-point boundary value problems

- Mathematics, Computer ScienceComputing
- 2005

A method for proving the existence of a solution within a “close”C1-neighborhood of an approximate solution of nonlinear two-point boundary value problems with Sturm-Liouville or periodic boundary operatorsB0 andB1 is presented.

### A Posteriori Error Bounds for Two-Point Boundary Value Problems

- Mathematics
- 1981

Consider a general two-point boundary value problem (TPBVP): \[ \begin{gathered} y'(t) = f(t,y), \hfill \\ B_1 y(a) + B_2 y(b) = w,\quad \hfill \\ \end{gathered} a \leqq t \leqq b, \] where $f:R^{n +…

### Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods

- Mathematics
- 1979

For a given starting point, the sequence of Newton iterates is well known to be invariant under affine transformation of the operator equation to be solved. This property, however, is not…

### Verification of Positive Definiteness

- Computer Science, Mathematics
- 2006

A computational, simple and fast sufficient criterion to verify positive definiteness of a symmetric or Hermitian matrix is presented, based on a floating-point Cholesky decomposition and improves a known result.