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Introduction to Linear Algebra with Mathematica

Glossary

Parabolic Equations

We conside a general initial boundary value porlem for general linear heat equation with Dirichlet boundary conditions:
\begin{align} \label{eq:heateq} u_t &= \alpha u_{xx} - \gamma u + f(x,t) & x>0, \quad 00 &&\text{Initial condition} \\ \label{eq:regcondu_0} u_0,u_0' &\in L^1(\R_{\geq 0}) & &&\text{Regularity condition on $u_0, u_0'$} \\ \label{eq:regcondg} g &\in L^1([0,t]) & 0\leq t<\infty &&\text{Regularity condition on $g$} \\ \label{eq:regcondf} \|f(0,\cdot)\|_{L^{\infty}([0,t])} &, \: \|f_x(\cdot,t)\|_{L^1(\mathbb{R}_{\geq0})}, \: \sup_{\eta \geq0} \|f(\cdot,\eta)\|_{L^1(\R_{\geq 0})} < \infty & 0\leq t<\infty &&\text{Regularity conditions on $f$} \end{align}

 

  1. Boyd, J.P. and Flyer, N., Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods, Computer Methods in Applied Mechanics and Engineering, 1999, Volume 175, Issues 3–4, Pages 281--309. https://doi.org/10.1016/S0045-7825(98)00358-2
  2. Davis, C.-I. R. and Fornberg, B., A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs, Complex Variables and Elliptic Equations, 2014, Vol. 59, No. 4, pp. 564--577. doi: 10.1080/17476933.2013.766883
  3. Decuninck, B., Trogdon, T., Vasan, V., The method of Fokas for solving linear partial differential equations, SIAM Review, 56, 1 (2014), 159-186.
  4. Fokas, A.S., A unified transform method for solving linear and certain nonlinear PDEs, Proceedings of the Royal Society. Ser. A: Mathematical, Physical and Engineering Sciences, 453 (1997), 1411-1443.
  5. Fokas, A.S., On the integrability of linear and nonlinear partial differential equations, Journal of Mathematical Physics, 41, 6 (2000), 4188-4237.
  6. Fokas, A.S., A new transform method for evolution PDEs, IMA Journal of Applied Mathematics, 67, 6 (2002), 559-590.
  7. Reference list of publications on Fokas method/unified transform method.
  8. Fokas, A. S. and Gelfand, I. M., Surfaces on Lie groups, on Lie algebras, and their integrability, Communications in Mathematical Physics, 1996, Vol. 177, no. 1, 203--220. https://projecteuclid.org/euclid.cmp/1104286243
  9. Fokas, A.S., Pelloni, B., Unified Transform for Boundary Value Problems, CBMS-SIAM, 2008.
  10. Hashemzadeh, P., Fokas, A. S., Smitheman, S.A., A numerical technique for linear elliptic partial differential equations in polygonal domains, Proceedings of the Royal Society A, 2015, 741, 20140747; https://doi.org/10.1098/rspa.2014.0747
  11. Lopatinski Ya. B., A method of reduction of boundary-value problems for systems of differential equations of elliptic type to a system of regular integral equations (in Russian), Ukrainski Mathematical Journal, 1953, Vol. 5, pp. 123--151.
  12. Shapiro Z. Ja.,On general boundary problems for equations of elliptic type (in Russian), Izvestiya Akademii Nauk SSSR, Ser Mat, 1953, Vol. 17, pp. 539--562.

 

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