Theory and Modern Applications
Table 1 Ternary-FDT for some basic functions
From: Ternary-fractional differential transform schema: theory and application
Function w(x,y,t) | Transformed form |
|---|---|
\(t^{n\alpha _{1}}x^{m\alpha _{2}}y^{l\alpha _{3}}\) |
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\(E_{\alpha _{2}} (\lambda x^{\alpha _{2}} )={\sum_{j=0}^{\infty }\frac{\lambda ^{j}x^{j\alpha _{2} }}{\varGamma (j\alpha _{2}+1)}}\) |
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\(t^{n\alpha _{1}}x^{m\alpha _{2}}y^{l\alpha _{3}}E_{\alpha _{2}} (\lambda x^{\alpha _{2}} )\) |
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\(E_{\alpha _{1}} (\mu t^{\alpha _{1}} )E_{\alpha _{2}} (\lambda x^{\alpha _{2}} )E_{\alpha _{3}} (\nu y^{\alpha _{3}} )\) |
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\(\sin _{\alpha _{2}} (\lambda x^{\alpha _{2}} )={\sum_{j=0}^{\infty }\frac{(-1)^{j} (\lambda x^{\alpha _{2}} )^{2j+1}}{\varGamma ((2j+1)\alpha _{2}+1 )}}\) |
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\(\cos _{\alpha _{2}} (\lambda x^{\alpha _{2}} )={\sum_{j=0}^{\infty }\frac{(-1)^{j} (\lambda x^{\alpha _{2}} )^{2j}}{\varGamma (2j\alpha _{2}+1 )}}\) |
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\(\sinh _{\alpha _{2}} (\lambda x^{\alpha _{2}} )={\sum_{j=0}^{\infty }\frac{ (\lambda x^{\alpha _{2}} )^{2j+1}}{\varGamma ((2j+1)\alpha _{2}+1 )}}\) |
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\(\cosh _{\alpha _{2}} (\lambda x^{\alpha _{2}} )={\sum_{j=0}^{\infty }\frac{ (\lambda x^{\alpha _{2}} )^{2j}}{\varGamma (2j\alpha _{2}+1 )}}\) |
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