usfkad6_05.cpp

Solutions are obtained for Poissson, diffusion, or wave PDEs homogeneous or nonhomogeneous equations

if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="\\frac{x}{\\alpha_{x=X}X+1}f_{x=X}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}" + u + 
"& \\mbox{ if } " + K + "= 0;\\\\ \\sin " + K + " " + u + " &\\mbox{ otherwise } . \\end{array} \\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}\\frac{1}{\\alpha_" + U + " " + U +
" +1} & \\mbox{ if } " + K + " = 0;\\\\ \\frac{1}{\\alpha_" + U + "\\sin " + K + " "
+ U + " + " + K + " \\cos " + K + " " + U + "} & \\mbox{ otherwise } . \\end{array} \\right.  ";
symb1 = "";
symb2 = "";
}
sulmarker = 1;
}

//23 
if (f=="CCDRNH") 
{ 
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}1 + \\alpha_" + U + 
" (" + U + " - " + u + ") \\mbox{  if } " + K + " = 0; \\mbox{ otherwise} &\\mbox{}\\\\"
 + K + " \\cosh " + K + "("+U+"-" + u + ") + \\alpha_{"+U+"} \\sinh " + K + "("+U+"-" + u + ")  &\\mbox{} \\end{array}\\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}\\frac{1}{1 +\\alpha_" + U + "" + U
+ "} & \\mbox{ if } " + K + " = 0;\\\\ \\frac{1}{" + K + " \\cosh" + K + " " + U + 
"+ \\alpha_" + U + " \\sinh " + K + " " + U + " } & \\mbox{ otherwise} . \\end{array}\\right.   ";
symb1 = "";
symb2 = "";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="\\left(1-\\frac{\\alpha_{x=X}x}{\\alpha_{x=X}X+1}\\right)f_{x=0}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}1 + \\alpha_" + U + 
" (" + U + " - " + u + ") \\mbox{  if } " + K + " = 0; \\mbox{ otherwise} &\\mbox{}\\\\"
 + K + " \\cos " + K + "("+U+"-" + u + ") + \\alpha_{"+U+"} \\sin " + K + "("+U+"-" + u + ")  &\\mbox{} \\end{array}\\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}\\frac{1}{1 +\\alpha_" + U + "" + U
+ "} & \\mbox{ if } " + K + " = 0;\\\\ \\frac{1}{" + K + " \\cos" + K + " " + U + 
"+ \\alpha_" + U + " \\sin " + K + " " + U + " } & \\mbox{ otherwise} . \\end{array}\\right.   ";
symb1 = "";
symb2 = "";
}
sulmarker = 1;
}

//60
if(f=="CCDSNH")
{
ans1 = "e^{- " + K + " " + u + "}";
ans2 = "";
ans3 = "";
ans4 = "";
ans5="";
symb1="";
symb2="";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="f_{x=0}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";}
sulmarker = 1;
}

//145   Outgoing Diriclet wave, rectangular
if ( f=="CCDONH")
{
ans1 = "\\; e^{-i " + K + "\\," +u+ "}";
ans2="";
ans3="";
ans4="";
ans5="";
symb1="";
symb2="";
sulmarker = 1;
}

//146  Incoming Dirichlet wave, rectangular
if ( f=="CCDINH")
{
ans1 = "\\; e^{i " + K + "\\," +u+ "}";
ans2="";
ans3="";
ans4="";
ans5="";
symb1="";
symb2="";
sulmarker = 1;
}


//         CONSTANT COEFFICIENT EQUATION: NEUMANN CONDITION AT THE LOW END
//                    EIGENFUNCTIONS

//3
if (f=="CCNDHH")
{
ans1 = "\\cos \\kappa_" + u + " " + u + ""; //changed last u 
ans2 = " \\kappa_" + u + "=\\frac{\\pi}{2" + U + "},\\frac{3\\pi}{2" + U + "},\\frac{5\\pi}{2" + U + "}, \\ldots  ";
ans3 = "\\frac{2}{" + U + "}";
ans4 = "";
ans5 = ans1;
symb1 = "\\sum_{\\kappa_" + u + "}";
symb2 = "\\int_{0}^{" + U + "}\\,d" + u;
sulmarker = 1;
}

//4
if (f=="CCNNHH")
{
ans1="\\cos \\kappa_" + u + " " + u + "";
ans2=" \\kappa_" + u + " = 0, \\frac{\\pi}{" + U + "},\\frac{2\\pi}{" + U + "},\\frac{3\\pi}{" + U + "}, \\ldots  ";
ans3="N_{\\kappa_" + u + "}";
ans4=" N_{\\kappa_" + u + "} = \\left\\{\\begin{array}{ll}\\frac{1}{" + U 
+ "} & \\mbox{ if }\\kappa_" + u + " = 0; \\\\ \\frac{2}{" + U + "} & \\mbox{ otherwise} \\end{array} \\right.  "; //new line
ans5=ans1;
symb1 = "\\sum_{\\kappa_" + u + "}";
symb2 = "\\int_{0}^{" + U + "}\\,d" + u;
sulmarker = 1;
}


//7
if (f=="CCNRHH")
{
ans1="\\cos \\kappa_" + u + " " + u + "";
ans2="\\\\  \\{\\kappa_" + u + "\\} \\mbox{ are the solutions (maybe imaginary) to } \\\\   \\alpha_" 
+ U + " \\cos \\kappa_" + u + U + " -  \\sin \\kappa_" + u + U +" = 0.  \\\\";
ans3="\\frac{2}{" + U + " + \\frac{\\alpha_" + U + " }{\\alpha_" + U + "^2+\\kappa_" + u + "^2}}";
ans4="";
ans5=ans1;
symb1="\\sum_{\\kappa_" + u + "}";
symb2="\\int_{0}^{" + U + "}\\,d" + u + " ";
sulmarker = 1;
}

//56
if(f=="CCNSHH")
{
ans1 ="\\cos  \\kappa_" + u + " " + u + "";
ans2= "";
ans3="\\frac{2}{\\pi}";
ans4 = "";
ans5="\\cos  \\kappa_" + u + " " + u + " "; 
symb1="\\int_0^{\\infty} \\,d \\kappa_" + u + ""; 
symb2="\\int_0^{\\infty} \\,d " + u + "";
sulmarker = 1;
}

//         CONSTANT COEFFICIENT EQUATION: NEUMANN CONDITION AT THE LOW END
//                    NONHOMOGENEOUS FACTORS

//14  //MAY WANT TO CHANGE THIS TO UNITY
if (f=="CCNDHN")
{
ans1 = "\\cosh " + K + u + "";
ans2 = "";
ans3 = "\\frac{1}{\\cosh " + K + U + "}";
ans4 = "";
symb1 = "";
symb2 = "";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="f_{x=X}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\cos " + K + u + "";
ans2 = "";
ans3 = "\\frac{1}{\\cos " + K + U + "}";
ans4 = "";
symb1 = "";
symb2 = "";
}
sulmarker = 1;
}

//13
if (f=="CCNDNH")
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}" + U + 
"- " + u + " & \\mbox{ if }" + K + " = 0; \\\\ \\sinh " + K + " (" + U + " - " + u + ")& \\mbox{ otherwise} . \\end{array} \\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}- 1 & \\mbox{ if }" + K + " = 0; \\\\ \\frac{- 1}{" + K + "\\cosh " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1="";
symb2="";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))     // 1 dim, time
{ans1="(x-X)f_{x=0}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}" + U + 
"- " + u + " & \\mbox{ if }" + K + " = 0; \\\\ \\sin " + K + " (" + U + " - " + u + ")& \\mbox{ otherwise} . \\end{array} \\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}- 1 & \\mbox{ if }" + K + " = 0; \\\\ \\frac{- 1}{" + K + "\\cos " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1="";
symb2="";
}
sulmarker = 1;
}

//16
if (f=="CCNNHN")
{
ans1 = "\\cosh " + K + " " + u + "";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K+ "} = \\left\\{ \\begin{array}{ll}0 & \\mbox{ if }" + K + " =0;\\\\ \\frac{ 1}{" + K + "\\sinh " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1 = "";
symb2 = "";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="Cannot\\, solve\\, steady\\, state\\, Neumann\\, problem";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\cos " + K + " " + u + "";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K+ "} = \\left\\{ \\begin{array}{ll}0 & \\mbox{ if }" + K + " =0;\\\\ \\frac{ -1}{" + K + "\\sin " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1 = "";
symb2 = "";
}
sulmarker = 1;
}

//17
if (f=="CCNNNH")
{
ans1 = "\\cosh " + K + " ( " + U + " - " + u + " ) ";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}0 & \\mbox{ if } " + K + " = 0;\\\\ \\frac{ 1}{" + K + "\\sinh " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1 = ""; 
symb2 = "";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="Cannot\\, solve\\, steady\\, state\\, Neumann\\, problem";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\cos " + K + " ( " + U + " - " + u + " ) ";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}0 & \\mbox{ if } " + K + " = 0;\\\\ \\frac{ -1}{" + K + "\\sin " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array} \\right.  ";
symb1 = ""; 
symb2 = "";
}
sulmarker = 1;
}

//20
if (f=="CCNRHN")
{
ans1 = "\\cosh " + K + u + "";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{"+ K + "} = \\left\\{ \\begin{array}{ll}\\frac{1}{\\alpha_" + U + "} &\\mbox { if } " + K 
+ " = 0; \\\\ \\frac{1}{\\alpha_" + U + "\\cosh " + K + " " + U + " + " +
K + " \\sinh " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array}\\right.  ";
symb1 = "";
symb2 = ""; 
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="f_{x=X}/\\alpha_{x=X}";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\cos " + K + u + "";
ans2 = "";
ans3 = "M_{" + K + "}";
ans4 = " M_{"+ K + "} = \\left\\{ \\begin{array}{ll}\\frac{1}{\\alpha_" + U + "} &\\mbox { if } " + K 
+ " = 0; \\\\ \\frac{1}{\\alpha_" + U + "\\cos " + K + " " + U + " - " +
K + " \\sin " + K + " " + U + "} & \\mbox{ otherwise} . \\end{array}\\right.  ";
symb1 = "";
symb2 = ""; 
}
sulmarker = 1;
}

//25  This was "corrected" Nov 28; check carefully
if (f=="CCNRNH")
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}1 + \\alpha_" + U + 
" (" + U + " - " + u + ") \\mbox{  if } " + K + " = 0; \\mbox{ otherwise} &\\mbox{}\\\\"
 + K + " \\cosh " + K + "("+U+"-" + u + ") + \\alpha_{"+U+"} \\sinh " + K + "("+U+"-" + u + ")  &\\mbox{} \\end{array}\\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}\\frac{- 1}{\\alpha_" + U + "}& \\mbox{ if } " + K + 
" = 0;\\\\ \\frac{- 1/" + K + "}{" + K + "\\sinh " + K + " " + U + 
"+ \\alpha_" + U + " \\cosh " + K + " " + U + " } & \\mbox{ otherwise} . \\end{array} \\right .  ";
symb1 = ""; 
symb2 = "" ;
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="f_{x=0}\\left(x-\\frac{1+\\alpha_{x=X}X}{\\alpha_{x=X}}\\right)";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";
}
if (eqtype[1] == 1 && eqtype[0]==5)  // 1 dim, freq
{
ans1 = "\\eta_" + u + " (" + u + " ; " + K + ")";
ans2 = " \\eta_" + u + "(" + u + " ; " + K + ") = \\left\\{ \\begin{array}{ll}1 + \\alpha_" + U + 
" (" + U + " - " + u + ") \\mbox{  if } " + K + " = 0; \\mbox{ otherwise} &\\mbox{}\\\\"
 + K + " \\cos " + K + "("+U+"-" + u + ") + \\alpha_{"+U+"} \\sin " + K + "("+U+"-" + u + ")  &\\mbox{} \\end{array}\\right.  ";
ans3 = "M_{" + K + "}";
ans4 = " M_{" + K + "} = \\left\\{ \\begin{array}{ll}\\frac{- 1}{\\alpha_" + U + "}& \\mbox{ if } " + K + 
" = 0;\\\\ \\frac{ 1/" + K + "}{" + K + "\\sin " + K + " " + U + 
"- \\alpha_" + U + " \\cos " + K + " " + U + " } & \\mbox{ otherwise} . \\end{array} \\right .  ";
symb1 = ""; 
symb2 = "" ;
}
sulmarker = 1;
}

//61
if(f=="CCNSNH")
{
ans1 = "e^{- " + K + " " + u + "}";
ans2 = "";
ans3 = "\\left(\\frac{-1}{" + K + "} \\right)";
ans4 = "";
ans5 = "";
symb1 = "";
symb2 = "";
if (eqtype[1] == 1 && (eqtype[0]==1 || eqtype[0]==3))  // 1 dim, time
{ans1="Indeterminant \\, steady-state \\; for \\, Neumann \\, problem.";
ans2="";
ans3="";
ans4="";
symb1="";
symb2="";}
sulmarker = 1;
}

//147   Outgoing Neumann wave, rectangular
if ( f=="CCNONH")
{
ans1 = "\\; e^{-i " + K + "\\," +u+ "}";
ans2="";
ans3="\\frac{i}{"+ K + "}";
ans4="";
ans5="";
symb1="";
symb2="";
sulmarker = 1;
}

//148   Incoming Neumann wave, rectangular
if ( f=="CCNINH")
{
ans1 = "\\; e^{i " + K + "\\," +u+ "}";
ans2="";
ans3="\\frac{-i}{"+ K + "}";
ans4="";
ans5="";
symb1="";
symb2="";
sulmarker = 1;
}

//          CONSTANT COEFFICIENT EQUATION: ROBIN CONDITION AT THE LOW END
//                          EIGENFUNCTIONS


//6
if (f=="CCRDHH")
{
ans1="\\; \\eta_" + u + " (" + u + "; \\kappa_" + u + ")";
ans2="\\\\ \\eta_" + u + "("+ u + "; \\kappa_" + u + ") = \\sin\\kappa_" + u + " (" + U + "-" + u + 
") \\mbox{ where } \\{\\kappa_" + u + 
"\\} \\mbox{ are the nonzero solutions (maybe imaginary) to } \\\\ \\alpha_{"+u+"=0} \\sin \\kappa_"
 + u + U + " - \\kappa_"+ u + " \\cos \\kappa_ " + u + U + " = 0; \\\\ \\mbox{also if }\\alpha_{"+u+"=0} " +
U + " -1 = 0 \\mbox{ include } \\kappa_" + u + " = 0 , \\eta_" + u + 
"(" + u + ";0 ) = " + U + " - " + u + " . \\\\";
ans3="N_{\\kappa_" + u + "}";
ans4=" N_{\\kappa_" + u + "} = \\left\\{ \\begin{array}{ll}\\frac{3}{" + U + "^3} & \\mbox{ if } \\kappa_" + 
u + " = 0; \\\\ \\frac{2}{" + U + " + \\frac{\\alpha_{"+u+"=0} - 1}{\\alpha_{"+u+"=0}^2 + \\kappa_" + u + 
"{}^2 }} & \\mbox{ otherwise} . \\end{array} \\right.  ";
ans5=ans1;
symb1="\\sum_{\\kappa_" + u + "}";
symb2="\\int_{0}^{" + U + "}\\,d" + u + " ";
sulmarker = 1;
}


//8
if (f=="CCRNHH")
{
ans1="\\cos \\kappa_" + u + " (" + U + " - " + u + ")";
ans2="\\\\ \\{\\kappa_" + u + "\\} \\mbox{ are the solutions (maybe imaginary) to } \\alpha_{"+u+"=0} \\cos \\kappa_" +
u + U + " + \\kappa_" + u + "\\sin \\kappa_" + u + U + " = 0. \\\\";
ans3="\\frac{2}{" + U + " -\\frac{\\alpha_{"+u+"=0}}{\\alpha_{"+u+"=0}^2 + \\kappa_" + u + "^2}}"; ans4=""; 
ans5=ans1; 
symb1="\\sum_{\\kappa_" + u + "}";
symb2="\\int_{0}^{" + U + "}\\,d" + u + " ";
sulmarker = 1;
}

//9
if (f=="CCRRHH")
{
ans1="\\; \\eta_" + u + " (" + u + " ; \\kappa_" + u + ")"; 
ans2="\\eta_" + u + "(" + u + " ; \\kappa_" + u + ") = \\sin[\\kappa_" + u + " "  + u + 
" + \\arctan(\\frac{- \\kappa_" + u + 
"}{\\alpha_{"+u+"=0}}) ] \\mbox{ where } \\{\\kappa_" + u + 
"\\} \\mbox{ are the nonzero solutions (maybe imaginary)} \\\\ \\mbox{to      } - \\alpha_" + U + 
" \\sin [\\kappa_" + u + U + " + \\arctan(\\frac{- \\kappa_" + u + 
"}{\\alpha_{"+u+"=0}}) ] = \\kappa_" + u + "\\cos [\\kappa_" + u + U + 
" +\\arctan(\\frac{- \\kappa_" + u + 
"}{\\alpha_{"+u+"=0}}) ]; \\\\ \\mbox{also if }\\alpha_" + U + 
" -\\alpha_" + U + " \\alpha_{"+u+"=0} " + U + " - \\alpha_{"+u+"=0} = 0 \\mbox{ include }\\kappa_"
+ u + " = 0 , \\eta_" + u + "(" + u + " ; 0) = 1 - \\alpha_{"+u+"=0} " + u + ". \\\\";
ans3="N_{\\kappa_" + u + "}";
ans4=" N_{\\kappa_" + u + "} = \\left\\{ \\begin{array}{ll}\\frac{3}{3 " + U + " -3\\alpha_{"+u+"=0}" + 
 U + "^2 + \\alpha_{"+u+"=0}^2 " + U + "^3} & \\mbox{ if } \\kappa_" + u + " = 0; \\\\ \\frac{2}{" + U + 
 " +\\frac{\\alpha_" + U + " }{\\alpha_" + U +
"{}^2 + \\kappa_" + u + "{}^2} - \\frac{\\alpha_{"+u+"=0}}{\\alpha_{"+u+"=0}^2 + \\kappa_" + u + "{}^2 }} & \\mbox{ otherwise} . \\end{array}\\right.  ";
ans5=ans1;
symb1="\\sum_{\\kappa_" + u + "}";
symb2="\\int_0^{" + U + "}\\,d" + u + " ";
sulmarker = 1;
} 

//58
if(f=="CCRSHH")
{
ans1 = "\\sin [\\kappa_" + u + " " + u + " - \\arctan \\frac{\\kappa_" + u 
+ "}{\\alpha_{"+u+"=0}}]";
ans2 = "( \\mbox{ if }\\alpha_{"+u+"=0} > 0, \\mbox{another term } e^{- \\alpha_{"+u+"=0}"+ u 
+ "} \\mbox{must be included and spectrum is mixed - see text.})\\\\";
ans3="\\frac{2}{\\pi}";
ans4="";
ans5=ans1;
symb1="\\int_{0}^{\\infty} \\,d \\kappa_" + u + ""; 
symb2="\\int_{0}^{\\infty} \\,d " + u + "";
sulmarker = 1;
}

//