{"id":16574879,"url":"https://github.com/matnoble/pde_coding","last_synced_at":"2026-04-21T04:32:19.185Z","repository":{"id":159135804,"uuid":"193111749","full_name":"MatNoble/PDE_coding","owner":"MatNoble","description":"FDM, 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unexpected eof while reading","robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":false,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":[],"created_at":"2024-10-11T21:45:55.995Z","updated_at":"2026-04-21T04:32:19.170Z","avatar_url":"https://github.com/MatNoble.png","language":"MATLAB","funding_links":[],"categories":[],"sub_categories":[],"readme":"## Some Results\n\n### Landau–Lifshitz equation\n\n\u003cimg src=\"/image/LLEmodel.png?raw=true\" width=\"600px\"\u003e\n\n\u003cimg src=\"/image/lle.gif?raw=true\" width=\"300px\"\u003e\n\n### Marker and Cell Methond for Stokes Equations\n\n\u003cimg src=\"/image/MACmodel.png?raw=true\" width=\"500px\"\u003e\n\n\u003cimg src=\"/image/MAC0.png?raw=true\" width=\"300px\"\u003e\n\n\u003cimg src=\"/image/MAC1.png?raw=true\" width=\"300px\"\u003e\n\n\u003cimg src=\"/image/puv.png?raw=true\" width=\"400px\"\u003e\n\n\u003cimg src=\"/image/MAC2.png?raw=true\" width=\"300px\"\u003e\n\n\u003cimg src=\"/image/MAC3.png?raw=true\" width=\"300px\"\u003e\n\n## Using FDM or FEM solve poisson and heat equations in 1D and 2D\n\n### Basic equations\npoisson equation: \n\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?-\\Delta\u0026space;u\u0026space;=\u0026space;f\" title=\"-\\Delta u = f\" /\u003e\u003c/div\u003e\n\n\nheat equation: \n\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?u_t\u0026space;-\u0026space;\\Delta\u0026space;u\u0026space;=\u0026space;f\" title=\"u_t - \\Delta u = f\" /\u003e\u003c/div\u003e\n\n\u003chr\u003e\n\n### Finite Difference Methods\n\u003e \n- Replacing the dervatives by finite differences(to do finite difference scheme)\n\n  \u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?u^k_t\u0026space;=\u0026space;\\frac{u^{k\u0026plus;1}-u^k}{\\Delta\u0026space;t}\" title=\"u^k_t = \\frac{u^{k+1}-u^k}{\\Delta t}\" /\u003e\u003c/div\u003e\n  \n  \u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?\\begin{aligned}\u0026space;u_{x\u0026space;x}\u0026space;\u0026\u0026space;\\approx\u0026space;\\frac{u\\left(x_{i-1},\u0026space;y_{j}\\right)-2\u0026space;u\\left(x_{i},\u0026space;y_{j}\\right)\u0026plus;u\\left(x_{i\u0026plus;1},\u0026space;y_{j}\\right)}{h_{x}^{2}}\u0026space;\\\\\u0026space;u_{y\u0026space;y}\u0026space;\u0026\u0026space;\\approx\u0026space;\\frac{u\\left(x_{i},\u0026space;y_{j-1}\\right)-2\u0026space;u\\left(x_{i},\u0026space;y_{j}\\right)\u0026plus;u\\left(x_{i},\u0026space;y_{j\u0026plus;1}\\right)}{h_{y}^{2}}\u0026space;\\end{aligned}\" title=\"\\begin{aligned} u_{x x} \u0026 \\approx \\frac{u\\left(x_{i-1}, y_{j}\\right)-2 u\\left(x_{i}, y_{j}\\right)+u\\left(x_{i+1}, y_{j}\\right)}{h_{x}^{2}} \\\\ u_{y y} \u0026 \\approx \\frac{u\\left(x_{i}, y_{j-1}\\right)-2 u\\left(x_{i}, y_{j}\\right)+u\\left(x_{i}, y_{j+1}\\right)}{h_{y}^{2}} \\end{aligned}\" /\u003e\u003c/div\u003e\n  \n- Taylor series(to do error estimate)\n\n  \u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?f(x)=\\frac{f\\left(x_{0}\\right)}{0\u0026space;!}\u0026plus;\\frac{f^{\\prime}\\left(x_{0}\\right)}{1\u0026space;!}\\left(x-x_{0}\\right)\u0026plus;\\ldots\u0026plus;\\frac{f^{(n)}\\left(x_{0}\\right)}{n\u0026space;!}\\left(x-x_{0}\\right)^{n}\u0026plus;R_{n}(x)\" title=\"f(x)=\\frac{f\\left(x_{0}\\right)}{0 !}+\\frac{f^{\\prime}\\left(x_{0}\\right)}{1 !}\\left(x-x_{0}\\right)+\\ldots+\\frac{f^{(n)}\\left(x_{0}\\right)}{n !}\\left(x-x_{0}\\right)^{n}+R_{n}(x)\" /\u003e\u003c/div\u003e\n  \n  \u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?R_{n}(x)=\\mathrm{o}\\left[\\left(x-x_{0}\\right)^{n}\\right]\" title=\"R_{n}(x)=\\mathrm{o}\\left[\\left(x-x_{0}\\right)^{n}\\right]\" /\u003e\u003c/div\u003e\n\n#### Example 1\n[for the 1D heat problem](https://github.com/MatNoble/PDE_coding/tree/master/FD/1D):\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?\\left\\{\\begin{array}{c}{u_{t}-\u0026space;u_{x\u0026space;x}=f(x,\u0026space;t)\\left(x_{1}\u003cx\u003cx_{2}\\right)(t\u003e0)}\u0026space;\\\\\u0026space;{u(x,\u0026space;0)=g_{t}(x)\\left(x_{1}\u003cx\u003cx_{2}\\right)}\u0026space;\\\\\u0026space;{u\\left(x_{1},\u0026space;t\\right)=g_{1}(t),\u0026space;u\\left(x_{2},\u0026space;t\\right)=g_{2}(t)(t\u003e0)}\\end{array}\\right.\" title=\"\\left\\{\\begin{array}{c}{u_{t}- u_{x x}=f(x, t)\\left(x_{1}\u003cx\u003cx_{2}\\right)(t\u003e0)} \\\\ {u(x, 0)=g_{t}(x)\\left(x_{1}\u003cx\u003cx_{2}\\right)} \\\\ {u\\left(x_{1}, t\\right)=g_{1}(t), u\\left(x_{2}, t\\right)=g_{2}(t)(t\u003e0)}\\end{array}\\right.\" /\u003e\u003c/div\u003e\n\nwith specific parameters:\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?0.1\u003cx\u003c\\pi,\\\u0026space;0\u003ct\u003c0.2\" title=\"0.1\u003cx\u003c\\pi,\\ 0\u003ct\u003c0.2\" /\u003e\u003c/div\u003e\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?u(x,\u0026space;0)=10\u0026space;\\frac{\\cos\u0026space;\\left(\\left(\\frac{x-x_{1}}{2}\\right)^{2}\\right)\u0026space;\\sin\u0026space;\\left(\\left(x-x_{2}\\right)^{3}\\right)}{x^{\\frac{4}{5}}}\\\\\" title=\"u(x, 0)=10 \\frac{\\cos \\left(\\left(\\frac{x-x_{1}}{2}\\right)^{2}\\right) \\sin \\left(\\left(x-x_{2}\\right)^{3}\\right)}{x^{\\frac{4}{5}}}\\\\\" /\u003e\u003c/div\u003e\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?u\\left(x_{1},\u0026space;t\\right)=u\\left(x_{2},\u0026space;t\\right)=0\" title=\"u\\left(x_{1}, t\\right)=u\\left(x_{2}, t\\right)=0\" /\u003e\u003c/div\u003e\n\u003cdiv align=center\u003e\u003cimg src=\"https://latex.codecogs.com/svg.latex?f(x,\u0026space;t)=0\" title=\"f(x, t)=0\" /\u003e\u003c/div\u003e\n\nhave the flowing result:\n\u003cdiv align=center\u003e\u003cimg width=\"500\" height=\"500\" src=\"https://github.com/MatNoble/PDE_coding/blob/master/image/FD.png\"/\u003e\u003c/div\u003e\n\n\u003chr\u003e\n\n### Finite Element Methods\n\u003e \n- Integration by part\n- Variational \n- Weak formula\n- Garlekin formula\n- Finite dimensional space approximates infinite dimensional space \n\n\u003chr\u003e\n\n### Approximate functions\nGibbs phenomenon\n\u003cdiv align=center\u003e\u003cimg width=\"500\" height=\"500\" src=\"https://github.com/MatNoble/PDE_coding/blob/master/image/Gibbsphenomenon.png\"/\u003e\u003c/div\u003e\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmatnoble%2Fpde_coding","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fmatnoble%2Fpde_coding","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fmatnoble%2Fpde_coding/lists"}