Boundary value problems for systems of linear evolution equations

Treharne, P. A.

doi:10.1093/imamat/69.6.539

Cited by 7 publications

(12 citation statements)

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“…In [16], Treharne and Fokas study a three-dimensional linear system on the half line, originating from a problem in elasticity. We discuss how their systems fits within the algorithm outlined in this paper.…”

Section: A Three-dimensional Example Due To Fokas and Treharnementioning

confidence: 99%

“…We discuss how their systems fits within the algorithm outlined in this paper. Although [16] is a seminal paper, as the first work where the UTM is considered for a system, the arguments presented there are specific to that system (the same can be said for the considerations in [9]) and do not highlight generic features of the UTM as applied to systems. In particular, we have emphasized the role played by expressions involving symmetric functions of the roots of the dispersion relations, allowing us to bypass the need for any branch cuts.…”

Section: A Three-dimensional Example Due To Fokas and Treharnementioning

confidence: 99%

“…In fact, to our knowledge, the application of the UTM to the standard wave equation presented in the appendix here is new, albeit not surprising. Systems of equations are examined in [9,13,15,14] and [16]. In [16], a problem from thermoelastic deformation is examined, while in [9] two water waverelated systems are treated.…”

Section: Introductionmentioning

confidence: 99%

“…Systems of equations are examined in [9,13,15,14] and [16]. In [16], a problem from thermoelastic deformation is examined, while in [9] two water waverelated systems are treated. In [15], the UTM is applied to the wave equation in a moving domain, while in [13,14] the free Klein-Gordon equation is examined.…”

Section: Introductionmentioning

confidence: 99%

“…In [15], the UTM is applied to the wave equation in a moving domain, while in [13,14] the free Klein-Gordon equation is examined. The treatment in these papers is non-generic in the sense that the different branches of the dispersion relations of the systems considered are either polynomial or rational in the wave number (as in [9,15]) or are conveniently parameterized to avoid radicals, as in [13,14,16]. In addition, all five papers start from a Lax pair and construct a Riemann-Hilbert problem almost exclusively, with almost no hints as to how to generalize the more accessible UTM for scalar systems based on the use of Green's Theorem.…”

Section: Introductionmentioning

confidence: 99%

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Fokas’s Unified Transform Method for linear systems

et al. 2017

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We demonstrate the use of the Unified Transform Method or Method of Fokas for boundary value problems for systems of constant-coefficient linear partial differential equations. We discuss how the apparent branch singularities typically appearing in the global relation are removable, allowing the method to proceed, in essence, as for scalar problems. We illustrate the use of the method with boundary value problems for the Klein-Gordon equation and the linearized Fitzhugh-Nagumo system. The case of wave equations is treated separately in an appendix.

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Section: A Three-dimensional Example Due To Fokas and Treharnementioning

confidence: 99%

Section: A Three-dimensional Example Due To Fokas and Treharnementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fokas’s Unified Transform Method for linear systems

et al. 2017

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Initial-boundary value problems for linear PDEs with variable coefficients

Treharne

Fokas

2007

Math. Proc. Camb. Phil. Soc.

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In this paper, the simultaneous identification of damping or anti-damping coefficient and initial value for some PDEs is considered. An identification algorithm is proposed based on the fact that the output of system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schrödinger equation with internal anti-damping, and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.

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The unified transform for linear, linearizable and integrable nonlinear partial differential equations

Fokas

Lillo

2014

Phys. Scr.

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So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.

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Boundary value problems for systems of linear evolution equations

Cited by 7 publications

References 0 publications

Fokas’s Unified Transform Method for linear systems

Fokas’s Unified Transform Method for linear systems

Initial-boundary value problems for linear PDEs with variable coefficients

The unified transform for linear, linearizable and integrable nonlinear partial differential equations

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