Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable

References

• Burgers , J . 1948 . A Mathematical Model Illustrating the Theory of Turbulence, Advances in Applied Mechanics , New York : Academic Press .
   Google Scholar
• Hopf , E . 1950 . The partial differential equation u t  + uu x  = u xx . Comm. Pure Appl. Math. , 3 : 201 – 230 .
   Web of Science ®Google Scholar
• Black , F and Scholes , M . 1973 . The pricing of options and corporate liabilities . J. Polit. Econ. , 81 ( 3 ) : 637 – 654 .
   Web of Science ®Google Scholar
• Merton , RC . 1971 . Optimum consumption and portfolio rules in a continuous-time model . J. Econom. Theory , 3 : 373 – 413 . Erratum: ibid. 6 (1973)
   Web of Science ®Google Scholar
• Merton , RC . 1973 . Theory of rational option pricing . Bell J. Econ. Manage. Sci. , 4 ( 1 ) : 141 – 183 .
   Google Scholar
• Henry , D . 1981 . Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics , New York : Springer-Verlag .
   Google Scholar
• Goldstein , JA . 1985 . Semigroups of Linear Operators and Applications , Oxford : Oxford University Press .
   Google Scholar
• Kovalevskaya , S . 1875 . Zur Theorie der partiellen Differentialgleichungen . J. Reine Angew. Math. , 80 : 1 – 32 .
   Google Scholar
• Khavinson , D and Shapiro , HS . 1994 . The heat equation and analytic continuation: Ivar Fredholm's first paper . Expo. Math. , 12 ( 1 ) : 79 – 95 .
   Google Scholar
• Lutz , DA , Miyake , M and Schäfke , R . 1999 . On the Borel summability of divergent solutions of the heat equation . Nagoya Math. J. , 154 : 1 – 29 .
   Web of Science ®Google Scholar
• Balser , W . 2007 . Formal power series solutions of the heat equation in one spatial variable , Zürich : in Differential Equations and Quantum Groups . Vol. 9, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schaefke, eds., IRMA Lectures in Mathematics & Theoritical Physics, European Mathematical Society, pp. 49–58
   Google Scholar
• Balser , W . 1999 . Divergent solutions of the heat equation: On an article of Lutz, Miyake and Schäfke . Pacific J. Math. , 188 ( 1 ) : 53 – 63 .
   Web of Science ®Google Scholar
• Balser , W . 2005 . Power Series Solutions of the Inhomogeneous Heat Equation , Proceedings of Recent Trends in Microlocal Analysis . Vol. 1412, Surikaisekikenkyusho Kokyuroken, RIMS, pp. 151–158
   Google Scholar
• Balser , W and Malek , S . 2004 . Formal Solutions of the Complex Heat Equation in Higher Spatial Dimensions , Proceedings of Global and Asymptotik Analysis of Differential Equations in the Complex Domain . Vol. 1367, Surikaisekikenkyusho Kokyuroken, RIMS, pp. 95–102
   Google Scholar
• Anastassiou , GA and Gal , SG . 2010 . Quantitative estimates in the overconvergence of some singular intergrals . Commun. Appl. Anal. , 14 ( 1 ) : 13 – 20 .
   Google Scholar
• Gal , CG , Gal , SG and Goldstein , JA . 2008 . Evolution equations with real time variable and complex spatial variables . Complex Variables Elliptic Eqns , 53 ( 8 ) : 753 – 774 .
   Web of Science ®Google Scholar
• Gal , CG , Gal , SG and Goldstein , JA . 2010 . Higher order heat and Laplace type equations with real time variable and complex spatial variable . Complex Variables Elliptic Eqns , 55 ( 4 ) : 357 – 373 .
   Web of Science ®Google Scholar
• Gal , CG , Gal , SG and Goldstein , JA . 2012 . Wave and telegraph equations with real time variable and complex spatial variables . Complex Variables Elliptic Eqns , 57 ( 1 ) : 91 – 109 .
   Web of Science ®Google Scholar
• Polácik , P and Sverák , V . 2008 . Zeros of complex caloric functions and singularities of complex viscous Burgers equation . J. Reine Angew. Math. , 616 : 205 – 217 .
   Web of Science ®Google Scholar
• Li , Lu . 2009 . Isolated singularities of the 1D complex viscous Burgers equation . J. Dyn. Differ. Eqns , 21 ( 4 ) : 623 – 630 .
   Web of Science ®Google Scholar
• Kohr , G and Mocanu , P . 2005 . “ Special Chapters of Complex Analysis (in Romanian) ” . In Babes-Bolyai , Cluj-Napoca : University Press .
   Google Scholar
• Reed , M and Simon , B . 1975 . “ Methods of Modern Mathematical Physics ” . In Fourier Analysis, Self-Adjointness , Vol. 2 , New York, London : Academic Press [Harcourt Brace Jovanovich, Publishers] .
   Google Scholar