Numerical existence and uniqueness proof for solutions of semilinear parabolic equations

Minamoto, Teruya

doi:10.1016/s0893-9659(01)80031-8

Cited by 5 publications

(6 citation statements)

References 4 publications

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“…From the fact that the invertibility of the linearlized operator for parabolic equation is always valid, some other Newton type method is studied in [12,13]. However, the numerical examples in those works are prototype problems and it is not clear whether they can be applied to more realistic problems.…”

Section: Parabolic Initial-boundary Value Problemsmentioning

confidence: 99%

“…Therefore, if the function c takes negative value, then the right-hand side of (16) becomes very large and it leads to an over-estimation of the inverse operator L −1 t . In [13], a weighted norm on the time-dependent Sobolev space is used, but the influence of the exponential dependency on T still remains.…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…Therefore, it should be not efficient to use such a norm for the actual implementation of our verification procedures based on the Newton method. In [12,13], some verification procedures are presented by using the theoretical estimates similar to that in [31], but the verification cost seems to be high provided that the corresponding elliptic part, .i.e., −νΔu + (b · ∇)u + cu, is not coercive. On the other hand, in the results by [1,32] on the numerical verifications of time-periodic solutions in one space dimension, they use the spectral method with explicit eigenvalues and corresponding eigen-functions for the linear part of concerned nonlinear equation.…”

Section: Introductionmentioning

confidence: 99%

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On verified computations of solutions for nonlinear parabolic problems

Nakao

2014

NOLTA

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Abstract:We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the numerical verification methods of solutions for elliptic problems which we have developed up to now. Next, under some fundamental procedures of verification for parabolic problems based on the fixed point theorem with Newton's method, we describe a summary of our methods including additional new technique which could yield some improvements. The main contents of the paper consist of the guaranteed a posteriori estimates for the linearized inverse operators of parabolic type. In order to confirm the effectiveness of our methods, we give some numerical examples for the guaranteed bounds of iverse operators as well as give some prototype results for numerical verification of solutions of nonlinear parabolic problems. Moreover, we will mention an extention of the present technique to the verification of time-periodic solutions.

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Section: Parabolic Initial-boundary Value Problemsmentioning

confidence: 99%

Section: A Priori Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On verified computations of solutions for nonlinear parabolic problems

Nakao

2014

NOLTA

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“…The concrete value C L 2 L 2 ,L 2 H 1 0 > 0 satisfying (2) can be calculated by the Gronwall inequality or other theoretical considerations (e.g., [16]), which we call the "a priori estimates." However, in general, C L 2 L 2 ,L 2 H 1 0 obtained by such a priori estimates is exponentially dependent on the length of the time interval J unless the corresponding elliptic part of the operator L t is coercive [4,5]. Thus a priori estimates often lead to an overestimate for the norm of L −1 t , which yields worse results for some purposes.…”

Section: Introductionmentioning

confidence: 99%

“…5. In this section we also show some prototype results of the numerical enclosure of solutions for nonlinear parabolic problems as an application of our method.…”

Section: Introductionmentioning

confidence: 99%

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

2013

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We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newtontype formulation, we derive an a posteriori estimate of the norm for the infinitedimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.

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