hi can someone please help me to prove that
\(\displaystyle \lim_{x \to \infty} u_x(x,t) = 0 \)
where
u (x, t) = ∫ ∞-∞Φ (x - y, t) g (y) dy
and
\(\displaystyle u_x(x,t)=-\frac{1}{2t\sqrt{4\pi t}}\int\limits_{-\infty}^{\infty}(x-y)u_0(y)e^{-\frac{(x-y)^2}{4t}}dy\)
with
Φ (x, t) = (1 / (√ (4πt)) exp (-x2 / 4t)
and g is continuous uniformly bounded non-negative function having a property g > 0 on (a, b)
and g = 0 otherwise. Here −∞ < a < b < ∞.
thans in advance
\(\displaystyle \lim_{x \to \infty} u_x(x,t) = 0 \)
where
u (x, t) = ∫ ∞-∞Φ (x - y, t) g (y) dy
and
\(\displaystyle u_x(x,t)=-\frac{1}{2t\sqrt{4\pi t}}\int\limits_{-\infty}^{\infty}(x-y)u_0(y)e^{-\frac{(x-y)^2}{4t}}dy\)
with
Φ (x, t) = (1 / (√ (4πt)) exp (-x2 / 4t)
and g is continuous uniformly bounded non-negative function having a property g > 0 on (a, b)
and g = 0 otherwise. Here −∞ < a < b < ∞.
thans in advance
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