Geodesic
Construction of Eigenfunctions of the Tricomi--Neumann Problem for Equations of Mixed Type with Characteristic Degeneration and Their Application
Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 76-87

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For the equation of mixed type $$ L_\alpha u\equiv u_{xx}+yu_{yy}+\alpha u_y+\lambda u=0, $$ where $0\alpha1$ and $\lambda$ is a complex parameter, we obtain eigenvalues in a special domain by the method of separation of variables and construct the system of corresponding eigenfunctions of the Tricomi–Neumann spectral problem. We construct the solution of the Tricomi–Neumann problem as a sum of biorthogonal series. 
@article{MZM_2003_74_1_a8,
     author = {K. B. Sabitov and S. L. Bibakova},
     title = {Construction of {Eigenfunctions} of the {Tricomi--Neumann} {Problem} for {Equations} of {Mixed} {Type} with {Characteristic} {Degeneration} and {Their} {Application}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {76--87},
     publisher = {mathdoc},
     volume = {74},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a8/}
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K. B. Sabitov; S. L. Bibakova. Construction of Eigenfunctions of the Tricomi--Neumann Problem for Equations of Mixed Type with Characteristic Degeneration and Their Application. Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 76-87. http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a8/