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Fundamental solution
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<h2 id="example">Example</h2> <p>Consider the following differential equation <i>Lf</i> = sin(<i>x</i>) with L = d 2 d x 2 . {\displaystyle L={\frac {d^{2}}{dx^{2}}}.} </p><p>The fundamental solutions can be obtained by solving <i>LF</i> = <i>δ</i>(<i>x</i>), explicitly, d 2 d x 2 F ( x ) = δ ( x ) . {\displaystyle {\frac {d^{2}}{dx^{2}}}F(x)=\delta (x)\,.} </p><p>Since for the unit step function (also known as the <a href="/facts/Heaviside_function/bgsUtxgP">Heaviside function</a>) H we have d d x H ( x ) = δ ( x ) , {\displaystyle {\frac {d}{dx}}H(x)=\delta (x)\,,} there is a solution d d x F ( x ) = H ( x ) + C . {\displaystyle {\frac {d}{dx}}F(x)=H(x)+C\,.} Here C is an arbitrary constant introduced by the integration. For convenience, set <i>C</i> = −1/2. </p><p>After integrating d F d x {\displaystyle {\frac {dF}{dx}}} and choosing the new integration constant as zero, one has F ( x ) = x H ( x ) − 1 2 x = 1 2 | x | . {\displaystyle F(x)=xH(x)-{\frac {1}{2}}x={\frac {1}{2}}|x|~.} </p> <h2 id="motivation">Motivation</h2> <p>Once the fundamental solution is found, it is straightforward to find a solution of the original equation, through <a href="/facts/Convolution/PVPrdz9J">convolution</a> of the fundamental solution and the desired right hand side. </p><p>Fundamental solutions also play an important role in the numerical solution of partial differential equations by the <a href="/facts/Boundary_element_method/FMIb4mLs">boundary element method</a>. </p> <h3>Application to the example</h3> <p>Consider the operator L and the differential equation mentioned in the example, d 2 d x 2 f ( x ) = sin ( x ) . {\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)=\sin(x)\,.} </p><p>We can find the solution f ( x ) {\displaystyle f(x)} of the original equation by <a href="/facts/Convolution/PVPrdz9J">convolution</a> (denoted by an asterisk) of the right-hand side sin ( x ) {\displaystyle \sin(x)} with the fundamental solution F ( x ) = 1 2 | x | {\textstyle F(x)={\frac {1}{2}}|x|} : f ( x ) = ( F ∗ sin ) ( x ) := ∫ − ∞ ∞ 1 2 | x − y | sin ( y ) d y . {\displaystyle f(x)=(F*\sin )(x):=\int _{-\infty }^{\infty }{\frac {1}{2}}|x-y|\sin(y)\,dy\,.} </p><p>This shows that some care must be taken when working with functions which do not have enough regularity (e.g. compact support, <i>L</i>1 integrability) since, we know that the desired solution is <i>f</i>(<i>x</i>) = −sin(<i>x</i>), while the above integral diverges for all x. The two expressions for f are, however, equal as distributions. </p> <h3>An example that more clearly works</h3> <p> d 2 d x 2 f ( x ) = I ( x ) , {\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)=I(x)\,,} where I is the <a href="/facts/Indicator_function/QEuh04NM">characteristic (indicator) function</a> of the unit interval [0,1]. In that case, it can be verified that the convolution of <i>I</i> with <i>F</i>(<i>x</i>) = |<i>x</i>|/2 is ( I ∗ F ) ( x ) = { 1 2 x 2 − 1 2 x + 1 4 , 0 ≤ x ≤ 1 | 1 2 x − 1 4 | , otherwise {\displaystyle (I*F)(x)={\begin{cases}{\frac {1}{2}}x^{2}-{\frac {1}{2}}x+{\frac {1}{4}},&0\leq x\leq 1\\|{\frac {1}{2}}x-{\frac {1}{4}}|,&{\text{otherwise}}\end{cases}}} which is a solution, i.e., has second derivative equal to I. </p> <h3>Proof that the convolution is a solution</h3> <p>Denote the <a href="/facts/Convolution/PVPrdz9J">convolution</a> of functions F and g as <i>F</i> ∗ <i>g</i>. Say we are trying to find the solution of <i>Lf</i> = <i>g</i>(<i>x</i>). We want to prove that <i>F</i> ∗ <i>g</i> is a solution of the previous equation, i.e. we want to prove that <i>L</i>(<i>F</i> ∗ <i>g</i>) = <i>g</i>. When applying the differential operator, L, to the convolution, it is known that L ( F ∗ g ) = ( L F ) ∗ g , {\displaystyle L(F*g)=(LF)*g\,,} provided L has constant coefficients. </p><p>If F is the fundamental solution, the right side of the equation reduces to δ ∗ g . {\displaystyle \delta *g~.} </p><p>But since the delta function is an <a href="/facts/Identity_element/9nss3xHY">identity element</a> for convolution, this is simply <i>g</i>(<i>x</i>). Summing up, L ( F ∗ g ) = ( L F ) ∗ g = δ ( x ) ∗ g ( x ) = ∫ − ∞ ∞ δ ( x − y ) g ( y ) d y = g ( x ) . {\displaystyle L(F*g)=(LF)*g=\delta (x)*g(x)=\int _{-\infty }^{\infty }\delta (x-y)g(y)\,dy=g(x)\,.} </p><p>Therefore, if F is the fundamental solution, the convolution <i>F</i> ∗ <i>g</i> is one solution of <i>Lf</i> = <i>g</i>(<i>x</i>). This does not mean that it is the only solution. Several solutions for different initial conditions can be found. </p> <h2 id="fundamental-solutions-for-some-partial-differential-equations">Fundamental solutions for some partial differential equations</h2> <p>The following can be obtained by means of Fourier transform: </p> <h3>Laplace equation</h3> <p>For the <a href="/facts/Laplace_equation/8n1YkmFS">Laplace equation</a>, [ − Δ ] Φ ( x , x ′ ) = δ ( x − x ′ ) {\displaystyle [-\Delta ]\Phi (\mathbf {x} ,\mathbf {x} ')=\delta (\mathbf {x} -\mathbf {x} ')} the fundamental solutions in two and three dimensions, respectively, are Φ 2D ( x , x ′ ) = − 1 2 π ln | x − x ′ | , Φ 3D ( x , x ′ ) = 1 4 π | x − x ′ | . {\displaystyle \Phi _{\textrm {2D}}(\mathbf {x} ,\mathbf {x} ')=-{\frac {1}{2\pi }}\ln |\mathbf {x} -\mathbf {x} '|,\qquad \Phi _{\textrm {3D}}(\mathbf {x} ,\mathbf {x} ')={\frac {1}{4\pi |\mathbf {x} -\mathbf {x} '|}}~.} </p> <h3>Screened Poisson equation</h3> <p>For the <a href="/facts/Screened_Poisson_equation/Le3hv9Wf">screened Poisson equation</a>, [ − Δ + k 2 ] Φ ( x , x ′ ) = δ ( x − x ′ ) , k ∈ R , {\displaystyle [-\Delta +k^{2}]\Phi (\mathbf {x} ,\mathbf {x} ')=\delta (\mathbf {x} -\mathbf {x} '),\quad k\in \mathbb {R} ,} the fundamental solutions are Φ 2D ( x , x ′ ) = 1 2 π K 0 ( k | x − x ′ | ) , Φ 3D ( x , x ′ ) = exp ( − k | x − x ′ | ) 4 π | x − x ′ | , {\displaystyle \Phi _{\textrm {2D}}(\mathbf {x} ,\mathbf {x} ')={\frac {1}{2\pi }}K_{0}(k|\mathbf {x} -\mathbf {x} '|),\qquad \Phi _{\textrm {3D}}(\mathbf {x} ,\mathbf {x} ')={\frac {\exp(-k|\mathbf {x} -\mathbf {x} '|)}{4\pi |\mathbf {x} -\mathbf {x} '|}},} where K 0 {\displaystyle K_{0}} is a <a href="/facts/Bessel_function/hSyvFPqC">modified Bessel function</a> of the second kind. </p><p>In higher dimensions the fundamental solution of the screened Poisson equation is given by the <a href="/facts/Bessel_potential/y159cSI1">Bessel potential</a>. </p> <h3>Biharmonic equation</h3> <p>For the <a href="/facts/Biharmonic_equation/2ECm99X8">Biharmonic equation</a>, [ − Δ 2 ] Φ ( x , x ′ ) = δ ( x − x ′ ) {\displaystyle [-\Delta ^{2}]\Phi (\mathbf {x} ,\mathbf {x} ')=\delta (\mathbf {x} -\mathbf {x} ')} the biharmonic equation has the fundamental solutions Φ 2D ( x , x ′ ) = − | x − x ′ | 2 8 π ln | x − x ′ | , Φ 3D ( x , x ′ ) = | x − x ′ | 8 π . {\displaystyle \Phi _{\textrm {2D}}(\mathbf {x} ,\mathbf {x} ')=-{\frac {|\mathbf {x} -\mathbf {x} '|^{2}}{8\pi }}\ln |\mathbf {x} -\mathbf {x} '|,\qquad \Phi _{\textrm {3D}}(\mathbf {x} ,\mathbf {x} ')={\frac {|\mathbf {x} -\mathbf {x} '|}{8\pi }}~.} </p> <h2 id="signal-processing">Signal processing</h2> <p class="note">Main article: <a href="/facts/Impulse_response/l4vaE5er">Impulse response</a></p> <p>In <a href="/facts/Signal_processing/Npaja6zb">signal processing</a>, the analog of the fundamental solution of a differential equation is called the <a href="/facts/Impulse_response/l4vaE5er">impulse response</a> of a filter. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Green%27s_function/rKxKKPcp">Green's function</a></li> <li><a href="/facts/Impulse_response/l4vaE5er">Impulse response</a></li> <li><a href="/facts/Parametrix/dT6DfH2g">Parametrix</a></li></ul> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Fundamental_solution">"Fundamental solution"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li>For adjustment to Green's function on the boundary see <a href="http://www.math.lsa.umich.edu/~sijue/greensfunction.pdf">Shijue Wu notes</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Smoller, Joel (1994). "7. Distribution Theory". Shock Waves and Reaction—Diffusion Equations (2 ed.). Springer New York. ISBN 978-0-387-94259-9. <a href="978-0-387-94259-9" target="_blank">978-0-387-94259-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>