Analytical Particular Solutions of Multiquadrics Associated with Polyharmonic Operators

Algorithm 1

Mathematica codes for obtaining 2D-PMQs.

𝙻=3;  (*set an arbitrary L*)

𝙵𝚓_∶=(−1)^𝙻−𝚓2^𝙻+𝚓(2𝚒+2𝙻−2𝚓−3)!!((𝚒+𝙻)!)²()!()𝙰𝚒+𝙻−𝚓𝚒!)(2𝚒−3!!𝙻,𝚓;

𝙶𝚓_∶=(−1)^𝙻−𝚓2^{𝙻+𝚓−1}(2𝚒+2𝙻−2𝚓−1)!!((𝚒+𝙻)!)²()!()𝙱𝚒+𝙻−𝚓𝚒+𝙻−𝚓)(𝚒!)(2𝚒−3!!𝙻,𝚓;

𝙵𝙶[𝚒_]∶=∑^𝙻_𝚓=0𝙵𝚓+∑^𝙻−1_𝚓=0𝙶𝚓;

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}];

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

Phi𝙻 =∑^𝙻_𝚓=0𝙰𝙰𝙻,𝚓𝚁^2𝚓1+𝚁²+∑^𝙻−1_𝚓=0𝙱𝙱𝙻,𝚓𝙻𝚘𝚐1+1+𝚁²2𝚁^2𝚓+∑^𝙻−1_𝚓=0𝙲𝙲𝙻,𝚓𝚁^2𝚓

Series[Phi𝙻,{𝚁,0,2𝙻+2}]

PhiTemp = Phi𝙻;(*Check the PMQ*)

Do𝜕𝙿𝚑𝚒0=^𝚁𝚁𝜕_𝚁𝙿𝚑𝚒𝚃𝚎𝚖𝚙𝚁;𝙿𝚑𝚒𝚃𝚎𝚖𝚙=𝙿𝚑𝚒0,{𝚓,1,𝙻};

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