List of Differential Equations Formulas Latex Code

rockingdingo 2023-01-28 #math #differential equations #laplace #bessel

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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

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

Latex Code

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

Explanation

$u_{t}$ : First Order Partial Derivative of function u(x,t) over time t.

$u_{xx}$ : Second Order Partial Derivative of function u(x,t) over x.

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Mathematical Formula Handbook

Derivation of the Diffusion Equation

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1.1 Wave Equation

Equation

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

Latex Code
```
            u_{tt}=c^{2}u_{xx}
        
```
Explanation
- $u_{tt}$ : Second Order Partial Derivative of function u(x,t) over time t.
- $u_{xx}$ : Second Order Partial Derivative of function u(x,t) over x.
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1.2 Heat Equation

Equation

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

Latex Code
```
            u_{t}={\alpha}^{2}u_{xx}
        
```
Explanation
- $u_{t}$ : First Order Partial Derivative of function u(x,t) over time t.
- $u_{xx}$ : Second Order Partial Derivative of function u(x,t) over x.
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1.3 Laplace Equation

Equation

$u_{xx} = 0$

Latex Code
```
            u_{xx} = 0
        
```
Explanation
- $u_{xx}$ : Second Order Partial Derivative of function u(x,t) over x.
Related Documents

1.4 Legendre's Equation

Equation

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

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 $P_{l}(x)=\frac{1}{2^{l}l!}(\frac{\mathrm{d}}{\mathrm{d} x})^{l}(x^2-1)^{l}$
Recursion relation: $P_{l}(x)=\frac{1}{l}[(2l-1)xP_{l-1}(x)-(l-1)P_{l-2}(x)]$

1.6 Spherical Harmonics Equation

Equation

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

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

1.0 Diffusion (conduction) Equation

Equation

Latex Code

Explanation

Related Documents

1.1 Wave Equation

Equation

Latex Code

Explanation

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1.2 Heat Equation

Equation

Latex Code

Explanation

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1.3 Laplace Equation

Equation

Latex Code

Explanation

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1.4 Legendre's Equation

Equation

Latex Code

Explanation

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1.5 Bessel's equation

Equation

Latex Code

Explanation

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1.6 Spherical Harmonics Equation

Equation

Latex Code

Explanation

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