A Note on the Triple Laplace Transform and Its Applications to Some Kind of Third-Order Differential Equation

Table 1

Table of triple Laplace transform for some function of three variables.

Functions𝑓(𝑥,𝑦,𝑡)	Triple laplace transform 𝐹(𝑝,𝑠,𝑘)
𝑎𝑏𝑐	𝑎𝑏𝑐𝑝𝑠𝑘
𝑥𝑦𝑡	1𝑝^2𝑠^2𝑘^2
𝑥^𝑛𝑦^𝑚𝑡^𝑣,𝑛,𝑚,𝑣 are natural numbers	𝑘^−1−𝑣𝑠^−1−𝑚𝑝^−𝑛−1Γ(1+𝑛)Γ(1+𝑚)Γ(1+𝑣)
𝑥^𝑛𝑦^𝑚𝑡^𝑣𝑒^−𝑎𝑥−𝑏𝑦−𝑐𝑡	(𝑘+𝑐)^−1−𝑣(𝑠+𝑏)^−1−𝑚𝑝+𝑎^−𝑛−1Γ(1+𝑛)Γ(1+𝑚)Γ(1+𝑣)
𝑒^−𝑎𝑥−𝑏𝑦−𝑐𝑡	1𝑎+𝑝(𝑏+𝑠)(𝑐+𝑘)
cos(𝑥)cos(𝑦)cos(𝑡)	𝑘𝑠𝑝1+𝑝^21+𝑠^21+𝑘^2
sin(𝑥)sin(𝑦)sin(𝑡)	11+𝑝^21+𝑠^21+𝑘^2
sin𝑥+𝑦+𝑡	−1+𝑝𝑠+𝑘𝑝+𝑠1+𝑝^21+𝑠^21+𝑘^2
cos(𝑥+𝑦+𝑡)	−𝑘+𝑝+𝑠−𝑘𝑝𝑠1+𝑝^21+𝑠^21+𝑘^2
𝑥𝑦𝑡	𝜋√𝜋8^3𝑘𝑠𝑝
𝑒^𝑎𝑥+𝑦𝑏+𝑐𝑡sinh(𝑎𝑥)sinh(𝑏𝑦)sinh(𝑐𝑡)	(𝑏)(𝑐)(𝑎)−2𝑎𝑝+𝑝^2−2𝑏𝑠+𝑠^2−2𝑐𝑘+𝑘^2
𝑒^𝑎𝑥+𝑦𝑏+𝑐𝑡cosh(𝑎𝑥)cosh(𝑏𝑦)cosh(𝑐𝑡)	(𝑏−𝑠)(𝑐−𝑘)𝑎−𝑝−2𝑎𝑝+𝑝^2−2𝑏𝑠+𝑠^2−2𝑐𝑘+𝑘^2
𝑎Erf2√𝑥𝑏Erf2√𝑦𝑐Erf2√𝑡	𝑒^−√𝑐2𝑘−𝑏2𝑠𝑘𝑝𝑠−1+𝑒^−√𝑐2𝑘1−𝑒^−𝑎2𝑝−1+𝑒^−𝑏2𝑠
sin(𝑎𝑥)𝑥sin(𝑏𝑦)𝑦sin(𝑐𝑡)𝑡	√arctan𝑎^2𝑝arctan𝑏^2𝑠√arctan𝑐^2𝑘
cos(𝑎𝑥)𝑥^𝑛cos(𝑏𝑦)𝑦^𝑚cos(𝑐𝑡)𝑡^𝑣	𝑘^−1+𝑣𝑏1+^2𝑠^21/2(−1+𝑚)𝑠^−1+𝑚cos[𝑐𝑡]cos(−1+𝑚)arctan𝑏^2𝑠Γ(1−𝑚)Γ(1−𝑣); ×𝑎1+^2𝑝^21/2(−1+𝑛)𝑝^−1+𝑛()|𝑎|cos𝑛−1arctan𝑝Γ(1−𝑛)
sin(𝑎𝑥)𝑥^𝑛sin(𝑏𝑦)𝑦^𝑚sin(𝑐𝑡)𝑡^𝑣	𝑘^−1+𝑣𝑏1+^2𝑠^21/2(−1+𝑚)𝑠^−1+𝑚Γ()Γ()𝑎1−𝑚1−𝑣1+𝑝^221/2(−1+𝑛); ×𝑝^−1+𝑛Γ()(𝑎)()|𝑎|1−𝑛signsin𝑛−1arctan𝑝Γ(1−𝑛)
𝐽𝑛(𝑥)𝐽𝑛𝑦𝐽𝑛(𝑡)	8^−𝑛𝑘𝑠𝑝^−1−𝑛Hypergeometric2𝐹11+𝑛2,2+𝑛21,−𝑘^2Hypergeometric2𝐹11+𝑛2,2+𝑛21,−𝑠^2Hypergeometric2𝐹11+𝑛2,2+𝑛21,−𝑝^2
𝐼𝑛(𝑥)𝐼𝑛𝑦𝐼𝑛(𝑡)	8^−𝑛𝑘𝑠𝑝^−1−𝑛Hypergeometric2𝐹11+𝑛2,2+𝑛21,1+𝑛,𝑘^2Hypergeometric2𝐹11+𝑛2,2+𝑛21,1+𝑛,𝑠^2Hypergeometric2𝐹11+𝑛2,2+𝑛21,1+𝑛,𝑝^2