id author title date pages extension mime words sentence flesch summary cache txt american_scientific_journal-4152 Baloch, Maria; Iqbal, Saleem; Sarwar, Farhana; Abdul Rehman, Abdul Rehman The Relationship of Fractional Laplace Transform with Fractional Fourier, Mellin and Sumudu Transforms 2019 6 .pdf application/pdf 1733 73 57 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2βˆ’πœŽπœŽ2+πœ”πœ”2βˆ’2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½βˆ’(𝜎𝜎+π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž = π‘’π‘’βˆ’ 𝑖𝑖 cot𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½οΏ½1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2+πœ”πœ”2οΏ½βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž = π‘’π‘’βˆ’ 𝑖𝑖 cot 𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½ οΏ½οΏ½ 1 βˆ’ 𝑖𝑖 cot𝛼𝛼 2πœ‹πœ‹ οΏ½ (πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼)𝑒𝑒 𝑖𝑖 cot 𝛼𝛼 2 �𝑑𝑑2+πœ”πœ”2οΏ½βˆ’(π‘–π‘–πœ”πœ”)𝑑𝑑 csc 𝛼𝛼𝑑𝑑𝑑𝑑 ∞ βˆ’βˆž οΏ½ (7) From (5) and (7) we get the result ℒ𝛼𝛼{πœ™πœ™(𝑑𝑑)} = π‘’π‘’βˆ’ 𝑖𝑖 cot𝛼𝛼 2 �𝜎𝜎2+2π‘–π‘–πœŽπœŽπœ”πœ”οΏ½β„±π›Όπ›Ό{πœ™πœ™(𝑑𝑑)π‘’π‘’βˆ’πœŽπœŽπ‘‘π‘‘ csc 𝛼𝛼} (8) Eq (8) shows the relationship of fractional Laplace transform with fractional Fourier transform and this relation will reduces to classical relation of Laplace transform and Fourier transform if we put 𝛼𝛼 = πœ‹πœ‹ 2 . This reduces to classical relation between Laplace transform Mellin transform for 𝛼𝛼 = πœ‹πœ‹ 2 . cache/american_scientific_journal-4152.pdf txt/american_scientific_journal-4152.txt