Transform Methods for Solving Partial Differential Equations

Duffy, Dean G.

doi:10.1201/9781420035148

Cited by 177 publications

(114 citation statements)

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“…Surveying the literature [40,44,45] it can be seen that the same technique cannot be applied to the kernel (43). Finding an exact inversion, then, appears unlikely.…”

Section: Laplace Inversion Of the Kernelsmentioning

confidence: 99%

Extension of the continuum time-dependent Hartree-Fock method to proton states

Pardi

Stevenson

2014

Phys. Rev. E

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This paper deals with the solution of the spherically symmetric time-dependent Hartree-Fock approximation applied to nuclear giant monopole resonances in the small amplitude regime. The problem is spatially unbounded as the resonance state is in the continuum. The practical requirement to perform the calculation in a finite-sized spatial region yields an artificial boundary, which is not present physically. The question of how to ensure the boundary does not interfere with the internal solution, while keeping the overall calculation time low is studied. Here we propose an absorbing boundary condition scheme to handle the conflict. The derivation, via a Laplace transform method, and implementation is described. An inverse Laplace transform required by the absorbing boundaries is calculated using a method of nonlinear least squares. The accuracy and efficiency of the scheme is tested and results presented to support the case that they are an effective way of handling the artificial boundary.

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“…Surveying the literature [40,44,45] it can be seen that the same technique cannot be applied to the kernel (43). Finding an exact inversion, then, appears unlikely.…”

Section: Laplace Inversion Of the Kernelsmentioning

confidence: 99%

Extension of the continuum time-dependent Hartree-Fock method to proton states

Pardi

Stevenson

2014

Phys. Rev. E

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“…Equation (14) is now a linear PDE, which allows the solution of many types of problems using transform methods [Duffy, 1994].…”

Section: Transformed Equationmentioning

confidence: 99%

Clean two‐ and three‐dimensional analytical solutions of Richards' equation for testing numerical solvers

Tracy

2006

Water Resources Research

101

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[1] This technical note derives clean analytical solutions of Richards' equation for threedimensional unsaturated groundwater flow. Clean means that the boundary conditions and steady state solutions are closed form expressions and the transient solutions have relatively simple additional Fourier series terms. Two-dimensional versions of these solutions are also given. The primary purpose for the solutions is to test linear and nonlinear solvers in finite difference/volume/element computer programs for accuracy and scalability using architectures ranging from PCs to parallel high-performance computers. This derivation starts from the quasi-linear assumption of relative hydraulic conductivity varying exponentially with pressure head and the separate approximation that relative hydraulic conductivity varies linearly with moisture content. This allows a transformation to be used to create a linear partial differential equation. Separation of variables and Fourier series are then used to obtain the final solution. Physically reasonable material properties are also used. A total of four solutions are given in this technical note (steady state and transient solutions for two different boundary conditions of the sample problem).

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“…The great advantage of using Green functions lies in the fact that, it allows for analysis of the influences of the initial conditions on the wave propagation. It is necessary to emphasise that our approach for deriving Green functions is different from the various methods available in the literature, which either use Driac delta function or a single orthogonality condition of the eigenfunctions [15][16][17]. The originality of the method is that, it combines the Ritz method and the variational principle.…”

Section: Introductionmentioning

confidence: 99%

Exact solution of the Mindlin–Herrmann model for longitudinal vibration of an isotropic rod

et al. 2015

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This paper presents a new approach to the problem of coupled longitudinal and transversal propagation of stress waves in an isotropic thick and elastic rod, based on the Mindlin-Herrmann theory. The novelty is that, the Hamilton vriational principle is used not only for derivation of the governing equations and set of natural boundary conditions, but also for obtaining the exact solution in terms Green functions directly from the Langrangian.The success of this approach is based on the existence of multiple orthogonalities of the eigenfuctions. The proposed method is much easier than the standard approach of building Green functions. A numerical example illustrates the method of finding eigenfrequencies and eigenfunctions for isotropic Mindlin-Herrmann rod.

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Transform Methods for Solving Partial Differential Equations

Cited by 177 publications

References 0 publications

Extension of the continuum time-dependent Hartree-Fock method to proton states

Extension of the continuum time-dependent Hartree-Fock method to proton states

Clean two‐ and three‐dimensional analytical solutions of Richards' equation for testing numerical solvers

Exact solution of the Mindlin–Herrmann model for longitudinal vibration of an isotropic rod

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