Numerical analysis of Fokas' unified method for linear elliptic PDEs

Ashton, A. C. L.; Crooks, Kevin

doi:10.1016/j.apnum.2015.06.003

Cited by 13 publications

(4 citation statements)

References 16 publications

(29 reference statements)

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“…The unified transform method is also well-suited to solving a wide variety of different boundary value problems: Dirichlet, Neumann, Robin, and even non-separable boundary value problems. Furthermore, the unified transform method gives rise to simple numerical techniques for computing solutions; the numerical aspects have been explored extensively in the literature, see for example [5,11,16].…”

Section: The Unified Transform Methodsmentioning

confidence: 99%

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Solving PDEs of fractional order using the unified transform method

Fernandez

Băleanu

Fokas

2018

Applied Mathematics and Computation

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Section: The Unified Transform Methodsmentioning

confidence: 99%

“…for all integers n, i.e. that |4n ± α| ≥ 1 for all integers n. In other words, α must lie in one of the intervals [1,3], [5,7], [9,11], etc. The domain D + can be described as follows, working from (31):…”

Section: Applications and Extensions 31 A Worked Examplementioning

confidence: 99%

Solving PDEs of fractional order using the unified transform method

Fernandez

Băleanu

Fokas

2018

Applied Mathematics and Computation

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“…Not only explicit formulae for the solutions are provided, but such formulae have also been observed numerically to be very efficient, e.g. [48], [125], [65], [68], [73], [6], [96], [36] and have led to the development of novel hybrid numerical-analytical techniques. A crucial quality of these formulae appears to be that they are uniformly well-behaved near the boundary.…”

Section: Introductionmentioning

confidence: 99%

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

Chatziafratis,

Grafakos,

Kamvissis

2024

Dynamics of Partial Differential Equations

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We study the inhomogeneous Airy partial differential equation (also called Stokes or linearized Korteweg-de Vries equation with a negative sign) on the half-line with generic initial and boundary data in a classical smooth setting, via the formula provided by the Fokas unified transform method for linear evolution equations. We first present a suitable decomposition of that formula in the complex plane in order to appropriately interpret various terms appearing in it, thus securing convergence in a strict sense. Writing the solution in an Ehrenpreis-Palamodov form, our analysis allows for rigorous a posteriori verification of the full initial-boundary-value problem and a thorough investigation of the behavior of the solution near the boundaries of the spatiotemporal domain. We prove that the integrals in this representation converge uniformly to prescribed values and the solution admits a smooth extension up to the boundary only under certain data compatibility conditions (with implications for well-posedness, control theory and efficient numerical computations). Importantly, based on this analysis, we perform an effective asymptotic study of far-field dynamics. This yields new explicit asymptotic formulae which characterize the properties of the solution in terms of (in)compatibilities of the data at the 'corner' of the quadrant. In particular, the asymptotic behavior of the solution is sensitive to perturbations of the data at the origin. In all cases, even assuming the initial data to belong to the Schwartz class, the solution loses this property at soon as time becomes positive. Hereby, we report on the discovery of a novel type of a long-range instability phenomenon for linear dispersive differential equations. Our ideas are extendable to other Airy-like and more general problems for dispersive evolution equations.

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“…Hence, over the last years, it has been realized that the global relation can be solved numerically, using various techniques. The first class of those methods involves the pointwise collocation of the global relation on the spectral plane, [33,53,55,59,82,86,87,88,90], while recently, a Galerkin approach for the global relation has been introduced, [8,9,10]. In this Chapter, the main focus lies on the approach using collocation-type techniques.…”

Section: Solution Of the Generalized Dtn Map Via Collocation-type Methodsmentioning

confidence: 99%

A study of computational and algorithmic methods for solving partial differential equations based on the unified transform method

Grylonakis¹,

Γρυλωνάκης²

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Στην παρούσα διδακτορική διατριβή αναπτύχθηκαν αριθμητικές τεχνικές για την επίλυση γραμμικών, ελλειπτικών μερικών διαφορικών εξισώσεων (Μ.Δ.Ε.) με βάση τη μέθοδο του ενοποιημένου μετασχηματισμού (ή μέθοδος του Φωκά). Συγκεκριμένα, προτάθηκε προσυντονισμένο σχήμα για την αριθμητική επίλυση της γενικευμένης απεικόνισης Dirichlet-to-Neumann (DtN). Επίσης, αναπτύχθηκαν μέθοδοι επαναληπτικού χωρικού βηματισμού με βάση ανοιχτές μεθόδους χρονικής ολοκλήρωσης για την επίλυση της Μ.Δ.Ε. στο εσωτερικό του υπολογιστικού χωρίου. Ακόμα, προτάθηκε τεχνική εκτίμησης σφάλματος με χρήση της εξίσωσης global relation. Εν συνεχεία, προτάθηκε κλάση συμπαγών μεθόδων χωρικού βηματισμού για την επίλυση της Μ.Δ.Ε. στο εσωτερικό υπολογιστικού χωρίου, όπου γίνεται χρήση είτε ανοιχτών είτε κλειστών μεθόδων χρονικής ολοκλήρωσης. Επιπρόσθετα, προτάθηκε κλάση παράλληλων τεχνικών ενοποιημένου μετασχηματισμού με βάση τη μέθοδο διαχωρισμού του χωρίου.

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Numerical analysis of Fokas' unified method for linear elliptic PDEs

Cited by 13 publications

References 16 publications

Solving PDEs of fractional order using the unified transform method

Solving PDEs of fractional order using the unified transform method

Long-range instability of linear evolution PDE on semi-bounded domains via the Fokas method

A study of computational and algorithmic methods for solving partial differential equations based on the unified transform method

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