Problem 46053. Construct finite difference approximations of derivatives

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f' = (f_{j+1} - f_j})/h\"/></w:customXmlPr><w:r><w:t>f\\prime=\\frac{f_{j+1}-f_j}{h}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf\"><w:r><w:t>use Taylor series</w:t></w:r></w:hyperlink><w:r><w:t>. In the example above, one can write</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ...\"/></w:customXmlPr><w:r><w:t> f_{j+1} = f_j + h f\\prime +  \\frac{h^2}{2}f\\prime\\prime +  \\frac{h^3}{6}f\\prime\\prime\\prime + \\ldots</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Then solving for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"f'\"/></w:customXmlPr><w:r><w:t>$f\\prime$</w:t></w:r></w:customXml><w:r><w:t> and neglecting terms of order </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"h^2\"/></w:customXmlPr><w:r><w:t>$h^2$</w:t></w:r></w:customXml><w:r><w:t> and higher gives</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>       </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"f' = (f_{j+1} - f_j)/h - hf''/2\"/></w:customXmlPr><w:r><w:t>f\\prime = \\frac{f_{j+1} - f_j}{h} - \\frac{h}{2}f\\prime\\prime</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Because the exponent on </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"h\"/></w:customXmlPr><w:r><w:t>h</w:t></w:r></w:customXml><w:r><w:t> in the last term is 1, the method is called a first order method.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that takes the order</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> of the derivative and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>terms</w:t></w:r><w:r><w:t> indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>n = 1</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>terms = [1 0]</w:t></w:r><w:r><w:t>, and</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ coeffs   = [1 -1]\n errOrder = 1\n errCoeff = -0.5;]]></w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 435.55px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 217.775px; transform-origin: 407px 217.775px; vertical-align: baseline; "><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 380.183px 7.91667px; transform-origin: 380.183px 7.91667px; unicode-bidi: normal; "><span style="">In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is</span></span></div><div style="block-size: 37.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.7917px; text-align: left; transform-origin: 384px 18.7917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f' = (f_{j+1} - f_j})/h" style="width: 84.5px; height: 37.5px;" width="84.5" height="37.5"></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 365.642px 7.91667px; transform-origin: 365.642px 7.91667px; unicode-bidi: normal; "><span style="">where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf"><span style=""><span style="">use Taylor series</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 116.3px 7.91667px; transform-origin: 116.3px 7.91667px; unicode-bidi: normal; "><span style="">. In the example above, one can write</span></span></div><div style="block-size: 36.0833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.0417px; text-align: left; transform-origin: 384px 18.0417px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f_{j+1} = f_j + hf' + h^2 f''/2 + h^3 f'''/6 + ..." style="width: 226.5px; height: 36px;" width="226.5" height="36"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51.7333px 7.91667px; transform-origin: 51.7333px 7.91667px; unicode-bidi: normal; "><span style="">Then solving for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAkCAYAAAAD3IPhAAABNklEQVRYhe2XXbGEMAxGj4c6wAAGVsEqwEEdrINrAQ1IwAMWqgEL3IcmU3Zm2W2ZBnjgPPET2kyS5gtwc3MDQAM8Mu0aKydaIAAj0AMz4L/YLsBk4chDFh/k3sv9DLgP9n/yvq/tiJNNZ1LYn7LZuPFNkPdtbWdevEflF5qiXPsiZlm8y7TXFFaPSicLL+SfjAmjqAziSMi0d2L/qah34Yl18iKlKKye+S+bdcSTVA3dtCelaFg9/1U7Jo1OT9FCXsc1Retlq7EditbLVmM7jIaUoqoFuQdt94tcn4oKXdWesZcRwxGgFI1K9RGgFFXdEnE0Q1W3RBzNUBnIFUdTJi7SX3QEMBmOStFh6vAj7Yj1sf7dUHE8XKXXI4IjHelTektLVOZATE84yxGlIabJc4GecnMp/gGjCl04ApTc8QAAAABJRU5ErkJggg==" alt="f'" style="width: 17.5px; height: 18px;" width="17.5" height="18"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 94.9083px 7.91667px; transform-origin: 94.9083px 7.91667px; unicode-bidi: normal; "><span style=""> and neglecting terms of order </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="h^2" style="width: 15.5px; height: 19.5px;" width="15.5" height="19.5"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 53.3px 7.91667px; transform-origin: 53.3px 7.91667px; unicode-bidi: normal; "><span style=""> and higher gives</span></span></div><div style="block-size: 38.5833px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19.2917px; text-align: left; transform-origin: 384px 19.2917px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.5917px 7.91667px; transform-origin: 13.5917px 7.91667px; unicode-bidi: normal; "><span style="">       </span></span><span style="vertical-align:-15px"><img src="data:image/png;base64,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" alt="f' = (f_{j+1} - f_j)/h - hf''/2" style="width: 130px; height: 38.5px;" width="130" height="38.5"></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 81.3px 7.91667px; transform-origin: 81.3px 7.91667px; unicode-bidi: normal; "><span style="">Because the exponent on </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">h</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.35px 7.91667px; transform-origin: 191.35px 7.91667px; unicode-bidi: normal; "><span style=""> in the last term is 1, the method is called a first order method.</span></span></div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.717px 7.91667px; transform-origin: 110.717px 7.91667px; unicode-bidi: normal; "><span style="">Write a function that takes the order</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91.7917px 7.91667px; transform-origin: 91.7917px 7.91667px; unicode-bidi: normal; "><span style=""> of the derivative and a vector</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">terms</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 130.3px 7.91667px; transform-origin: 130.3px 7.91667px; unicode-bidi: normal; "><span style=""> indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example,</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.91667px; transform-origin: 19.25px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">n = 1</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.91667px; transform-origin: 13.6167px 7.91667px; unicode-bidi: normal; "><span style=""> and</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; ">terms = [1 0]</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "><span style="">, and</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 61.3px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 30.65px; transform-origin: 404px 30.65px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 7.91667px; transform-origin: 69.3px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> coeffs   = [1 -1]</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 50.05px 7.91667px; transform-origin: 50.05px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> errOrder = 1</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.833333px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.833333px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.833333px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.833333px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 65.45px 7.91667px; transform-origin: 65.45px 7.91667px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; "> errCoeff = -0.5;</span></span></div></div></div></div>

In solving a differential equation with a finite-difference method, one computes derivatives with various combinations of the function's values at chosen grid points. For example, the forward difference formula for the first derivative is

f' = (f_{j+1} - f_j)/h

where j is the grid index and h is the spacing between points. The systematic approach for deriving such formulas is to <http://www2.math.umd.edu/~dlevy/classes/amsc466/lecture-notes/differentiation-chap.pdf use Taylor series>. In the example above, one can write

f_{j+1} = f_j + h f' + h^2 f''/2 + h^3 f'''/6 + ...

Then solving for f' and neglecting terms of order h^2 and higher gives

f' = (f_{j+1} - f_j)/h - h f''/2

Because the exponent on h in the last term is 1, the method is called a first order method.

Write a function that takes the order |n| of the derivative and a vector |terms| indicating the terms to use (based on the number of grid cells away from the point in question) and produces a vector of coefficients, the order of the error term, and the numerical coefficient of the error term. In the above example, |n = 1| and |terms = [1 0]|, and

coeffs   = [1 -1]
 errOrder = 1
 errCoeff = -0.5;

Solve

Solution Stats

2 Solutions
2 Solvers

Last Solution submitted on Jan 16, 2021

Last 200 Solutions

Problem Recent Solvers2

Problem Tags

derivative finite difference numerical methods

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 46053. Construct finite difference approximations of derivatives

Solution Stats

Last 200 Solutions

Problem Recent Solvers2

Suggested Problems

More from this Author148

Problem Tags

Community Treasure Hunt

Problem 46053. Construct finite difference approximations of derivatives

Solution Stats

Last 200 Solutions

Problem Recent Solvers2

Suggested Problems

More from this Author148

Problem Tags

Community Treasure Hunt

Players