Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

Algorithm 2

Mathematica codes for obtaining 3D-PMQs.

𝙻=3;  (*set an arbitrary L*)

𝙵[𝚓_]∶=(−1)^𝙻−𝚓2^−𝚓(2𝚒+2𝙻+1)!!()()!2𝚒+1!!𝚒+𝙻()!()𝚒+𝙻−𝚓2𝚒+2𝙻−2𝚓−3!!(2𝚒−3)!!𝙰[𝙻,𝚓];

𝙶[𝚓_]∶=

 (−1)^{𝙻−𝚓+1}2^−𝚓(2𝚒+2𝙻+1)!!()()()!2𝚒+1!!2𝚒+2𝙻−2𝚓+1𝚒+𝙻()!()𝚒+𝙻−𝚓2𝚒+2𝙻−2𝚓−1!!(2𝚒−3)!!𝙱[𝙻,𝚓];

𝙵𝙶[𝚒_]∶=∑^𝙻_𝚓=0𝙵[𝚓]+∑^𝙻−1_𝚓=0𝙶[𝚓];

𝚜𝚘𝚕=𝚂𝚘𝚕𝚟𝚎[𝚃𝚊𝚋𝚕𝚎[𝙵𝙶[𝚒]==1,{𝚒,1,2𝙻+1}],

  𝙹𝚘𝚒𝚗[𝚃𝚊𝚋𝚕𝚎[𝙰[𝙻,𝚓],{𝚓,0,𝙻}],𝚃𝚊𝚋𝚕𝚎[𝙱[𝙻,𝚓],{𝚓,0,𝙻−1}]]];

𝙳𝚘[𝙰𝙰[𝙻,𝚓]=(𝙰[𝙻,𝚓]/.𝚜𝚘𝚕)[[1]],{𝚓,0,𝙻}];

𝙳𝚘𝙱𝙱𝙻,𝚓=(𝙱[𝙻,𝚓]/.𝚜𝚘𝚕)[[1]],{𝚓,0,𝙻−1};

𝙲𝙲[𝙻_,𝚒_]∶=

 ^𝚒∑_𝚓=0(−1)^𝚒−𝚓(2𝚒−2𝚓−3)!!𝙰𝙰[𝙻,𝚓](𝚒−𝚓)!2^𝚒−𝚓+∑^𝚒_𝚓=0(−1)^{𝚒−𝚓−1}(2𝚒−2𝚓−1)!!𝙱𝙱[𝙻,𝚓](𝚒−𝚓)!2^𝚒−𝚓(2𝚒−2𝚓+1);

𝙿𝚑𝚒𝙻=∑^𝙻_𝚓=0𝙰𝙰𝙻,𝚓𝚁^2𝚓1+𝚁²+∑^𝙻−1_𝚓=0[𝚁]𝙱𝙱𝙻,𝚓𝙰𝚛𝚌𝚂𝚒𝚗𝚑𝚁𝚁^2𝚓+∑^𝙻−1_𝚓=0𝙲𝙲𝙻,𝚓𝚁^2𝚓

Series[𝙿𝚑𝚒𝙻,{𝚁,0,2𝙻+2}]

PhiTemp = PhiL; (*Check the PMQ*)

Do𝜕𝙿𝚑𝚒0=^𝚁𝚁²𝜕_𝚁𝙿𝚑𝚒𝚃𝚎𝚖𝚙𝚁²;𝙿𝚑𝚒𝚃𝚎𝚖𝚙=𝙿𝚑𝚒0,{𝚓,1,𝙻};

Print["Laplace𝙻  PhiL  = ",  FullSimplify[Phi0]]