A unified transform method for solving linear and certain nonlinear PDEs

1997 ◽  
Vol 453 (1962) ◽  
pp. 1411-1443 ◽  
Author(s):  
A. S. Fokas
Keyword(s):  
Nonlinear Pdes ◽  
Transform Method ◽  
Unified Transform Method

scholarly journals Solving PDEs of fractional order using the unified transform method

2018 ◽  
Vol 339 ◽  
pp. 738-749 ◽  
Author(s):  
Arran Fernandez ◽  
Dumitru Baleanu ◽  
Athanassios S. Fokas
Keyword(s):  
Fractional Order ◽  
Transform Method ◽  
Unified Transform Method

scholarly journals The unified transform method for the Sasa–Satsuma equation on the half-line

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130068 ◽  
Author(s):  
Jian Xu ◽  
Engui Fan
Keyword(s):  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Transform Method ◽  
Global Relation ◽  
Unified Transform Method ◽  
Initial Boundary Value ◽  
Dirichlet To Neumann Map

We implement the unified transform method to the initial-boundary value (IBV) problem of the Sasa–Satsuma equation on the half line. In addition to presenting the basic Riemann–Hilbert formalism, which linearizes this IBV problem, we also analyse the associated general Dirichlet to Neumann map using the so-called global relation.

scholarly journals Solving Wiener–Hopf problems without kernel factorization

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140304 ◽  
Author(s):  
Darren G. Crowdy ◽  
Elena Luca
Keyword(s):  
Fluid Mechanics ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Kernel Functions ◽  
New Method ◽  
Transform Method ◽  
New Approach ◽  
Mathematical Ideas ◽  
Analytical Functions ◽  
Unified Transform Method

A new approach to solving problems of Wiener–Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener–Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

Fokas’s Unified Transform Method for linear systems

2017 ◽  
Vol 76 (3) ◽  
pp. 463-488 ◽  
Author(s):  
Bernard Deconinck ◽  
Qi Guo ◽  
Eli Shlizerman ◽  
Vishal Vasan
Keyword(s):  
Linear Systems ◽  
Transform Method ◽  
Unified Transform Method

scholarly journals A new transform method II: the global relation and boundary-value problems in polar coordinates

2010 ◽  
Vol 466 (2120) ◽  
pp. 2283-2307 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Boundary Values ◽  
Transform Method ◽  
Global Relation ◽  
The Fourier Transform

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called ‘global relation’ (GR)—an equation involving transforms of the boundary values—is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

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