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Register a new userSummability of formal solutions for some generalized moment partial differential equations
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The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability results of certain family of generalized linear moment partial differential equations with variable coefficients.
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Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.
Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.
In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $mathbf{C}^2$ for integers $m,ngeq2$ and nonzero meromorphic functions $h(z_1,z_2),k(z_1,z_2)$ in $mathbf{C}^2$. Meromorphic solutions to associated partial differential equations are also studied.
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. In this study, we construct self-similar solutions of some model degenerate partial differential equations of the second, third, and fourth order. These self-similar solutions are expressed in terms of hypergeometric functions.
We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.
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