Arithmetic partial differential equations, I

On Arithmetic Partial Differential Equations

Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic partial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equations, we give a complete solution, and for others, we extend previously known results. For example, we determine under which conditions two cons...

On Some Conjectures about Arithmetic Partial Differential Equations

In this paper, we study the arithmetic partial differential equations x′p = ax n and x ′ p = a. We solve a conjecture of Haukkanen, Merikoski, and Tossavainen (HMT, in short) about the number of solutions (conjectured to be finite) of the equation x ′ p = ax n and improve a theorem of HMT about finding the solutions of the same equation. Furthermore, we also improve another theorem of HMT about...

global results on some nonlinear partial differential equations for direct and inverse problems

در این رساله به بررسی رفتار جواب های رده ای از معادلات دیفرانسیل با مشتقات جزیی در دامنه های کراندار می پردازیم . این معادلات به فرم نیم-خطی و غیر خطی برای مسایل مستقیم و معکوس مورد مطالعه قرار می گیرند . به ویژه، تاثیر شرایط مختلف فیزیکی را در مساله، نظیر وجود موانع و منابع، پراکندگی و چسبندگی در معادلات موج و گرما بررسی می کنیم و به دنبال شرایطی می گردیم که متضمن وجود سراسری یا عدم وجود سراسر...

Partial Differential Equations Midterm

1), find the explicit fundamental solution to the heat equation ∆u + b · ∇u − u t = 0 in R n × (0, ∞). (1) Letting G be what you find, show u 0 (x) = lim t→0 + R n G(x, t; y, 0)u 0 (y) dy, (2) when u 0 is bounded and continuous on R n. Taking the Fourier transform (with respect to x) of the equation gives −|ξ| 2 ˆ u + ib · ξ ˆ u − ˆ u t = 0 − |ξ| 2 + ib · ξ ˆ u = ˆ u t =⇒û = ce −(|ξ| 2 +ib·ξ)t ...

Stochastic Partial Differential Equations