List of Differential Equations Formulas Latex Code

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List of Differential Equations Formulas Latex Code

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In this blog, we will summarize the latex code for differential equations formulas, including diffusion (conduction) equation, wave equation, heat equation, laplace equation, Legendre's equation, Bessel's equation, Spherical Harmonics Equation, etc.

• 1. Differential Equations
• 1.0 Diffusion (conduction) Equation
• 1.1 Wave Equation
• 1.2 Heat Equation
• 1.3 Laplace Equation
• 1.4 Legendre's Equation
• 1.5 Bessel's Equation
• 1.6 Spherical Harmonics Equation

1. Differential Equations

• 1.0 Diffusion (conduction) Equation

  Equation

  
  
  

  Latex Code

              \kappa u_{xx} = u_{t} \\
              \frac{\partial \Psi}{\partial t}=\kappa \triangledown^{2} \Psi
          

  Explanation

  • : First Order Partial Derivative of function u(x,t) over time t.
  • : Second Order Partial Derivative of function u(x,t) over x.

  Related Documents

  • Mathematical Formula Handbook
  • Derivation of the Diffusion Equation
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• 1.1 Wave Equation

  Equation

  
  
  

  Latex Code

              u_{tt}=c^{2}u_{xx}
          

  Explanation

  • : Second Order Partial Derivative of function u(x,t) over time t.
  • : Second Order Partial Derivative of function u(x,t) over x.

  Related Documents

  • Mathematical Formula Handbook
  • Wave Equation
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• 1.2 Heat Equation

  Equation

  
  
  

  Latex Code

              u_{t}={\alpha}^{2}u_{xx}
          

  Explanation

  • : First Order Partial Derivative of function u(x,t) over time t.
  • : Second Order Partial Derivative of function u(x,t) over x.

  Related Documents

  • Mathematical Formula Handbook
  • Heat Equation
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• 1.3 Laplace Equation

  Equation

  
  
  

  Latex Code

              u_{xx} = 0
          

  Explanation

  • : Second Order Partial Derivative of function u(x,t) over x.

  Related Documents

  • Mathematical Formula Handbook
  • Laplace Equation
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• 1.4 Legendre's Equation

  Equation

  
  
  

  Latex Code

              (1-x^{2})\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}-2x\frac{\mathrm{d} y}{\mathrm{d} x}+l(l+1)y=0 \\
  P_{l}(x)=\frac{1}{2^{l}l!}(\frac{\mathrm{d}}{\mathrm{d} x})^{l}(x^2-1)^{l}\\
  P_{l}(x)=\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]
          

  Explanation

  • Solutions of Legendre equations are Legendre polynomials
  • Recursion relation:

  Related Documents

  • Mathematical Formula Handbook
  • Heat Equation
  • ä¸?大å??å¾®å??æ?¹ç¨?ä¹?æ³¢æ?¹ç¨?(1)

• 1.5 Bessel's equation

  Equation

  
  
  

  Latex Code

              x^{2}\frac{\mathrm{d}^{2} y}{\mathrm{d} x^{2}}+x\frac{\mathrm{d} y}{\mathrm{d} x}+(x^{2}-m^{2})y=0 \\
              J_{m}(x)=\sum^{\infty}_{k=0}\frac{(-1)^{k} (x/2)^{m+2k}}{k!(m+k)!}
          

  Explanation

  • Solutions of Bessel equations are Bessel functions .

  Related Documents

  • Mathematical Formula Handbook
  • Brown Univ.: MATHEMATICA TUTORIAL for the First Course. Part V: Bessel equation

• 1.6 Spherical Harmonics Equation

  Equation

  
  
  

  Latex Code

              [\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}(\sin \theta \frac{\partial}{\partial \theta}) + \frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial \phi^{2}}) ] Y^{m}_{l} + l(l+1) Y^{m}_{l}=0 \\ Y^{m}_{l}(\theta,\phi)=\sqrt{\frac{2l+1}{4 \pi} \frac{(l-|m|)!}{(l+|m|)!}}P^{m}_{l}(\cos \theta) e^{im \phi} \times \begin{cases}(-1)^{m} & m\ge 0 \\ 1 & m <0 \end{cases}
          

  Explanation

  • Spherical Harmonics Equation
  • 
  • Spherical Harmonics Solution :
  • 

  Related Documents

  • Mathematical Formula Handbook
  • Spherical Harmonics Wiki
  • ç??è°?å?½æ?°ä»?ç»?ï¼?Spherical Harmonics)