On the Elementary Solution for the Partial Differential Operator $\circledcirc_c^{k}$ Related to the Wave Equation
2018 ◽
Vol 11
(2)
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pp. 390-399
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In this article, we defined the operator $\diamondsuit _{m,c}^{k}$ which is iterated $k$-times and is defined by$$\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{p}\frac{\partial ^{2}}{\partial x_{i}^{2}} +\frac{m^{2}}{2}\right)^{2} - \left(\sum_{j=p+1}^{p+q}\frac{\partial ^{2}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},$$where $m$ is a nonnegative real number, $c$ is a positive real number and $p+q=n$ is the dimension of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, $x=(x_{1},\ldots x_{n})\in\mathbb{R}^{n}$ and $k$ is a nonnegative integer. We obtain a causal and anticausal solutionof the operator $\diamondsuit _{m,c}^{k}$, iterated $k$-times.
2021 ◽
Vol 14
(3)
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pp. 881-894
1970 ◽
Vol 13
(1)
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pp. 1-7
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2018 ◽
Vol 7
(1)
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pp. 77-83
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2014 ◽
Vol 16
(04)
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pp. 1350046
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1989 ◽
Vol 26
(01)
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pp. 103-112
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2003 ◽
Vol 2003
(3)
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pp. 153-158
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