Volume 5, Issue 2
Exact Difference Schemes for Parabolic Equations

Int. J. Numer. Anal. Mod., 5 (2008), pp. 303-319.

Published online: 2008-05

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The Cauchy problem for the parabolic equation $$\frac{∂u}{∂t} =\frac{∂}{∂x} (k(x,t) \frac{∂u}{∂x}) + f(u,x,,t), x \in R, t > 0,$$ $$u(x,0) = u_0(x), x\in R,$$ is considered. Under conditions $u(x, t) = X(x)T_1(t) + T_2 (t)$, $\frac{∂u}{∂x} \neq 0$, $k(x,t)=k_1(x)k_2(t)$, $f(u,x,t) = f_1(x,t)f_2(u)$, it is shown that the above problem is equivalent to a system of two first-order ordinary differential equations for which exact difference schemes with special Steklov averaging and difference schemes with any order of approximation are constructed on the moving mesh. On the basis of this approach, the exact difference schemes are constructed also for boundary-value problems and multi-dimensional problems. Presented numerical experiments confirm the theoretical results investigated in the paper.

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@Article{IJNAM-5-303, author = {Lapinska-Chrzczonowicz , Magdalena and Matus , Piotr}, title = {Exact Difference Schemes for Parabolic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {2}, pages = {303--319}, abstract = {

The Cauchy problem for the parabolic equation $$\frac{∂u}{∂t} =\frac{∂}{∂x} (k(x,t) \frac{∂u}{∂x}) + f(u,x,,t), x \in R, t > 0,$$ $$u(x,0) = u_0(x), x\in R,$$ is considered. Under conditions $u(x, t) = X(x)T_1(t) + T_2 (t)$, $\frac{∂u}{∂x} \neq 0$, $k(x,t)=k_1(x)k_2(t)$, $f(u,x,t) = f_1(x,t)f_2(u)$, it is shown that the above problem is equivalent to a system of two first-order ordinary differential equations for which exact difference schemes with special Steklov averaging and difference schemes with any order of approximation are constructed on the moving mesh. On the basis of this approach, the exact difference schemes are constructed also for boundary-value problems and multi-dimensional problems. Presented numerical experiments confirm the theoretical results investigated in the paper.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/813.html} }
TY - JOUR T1 - Exact Difference Schemes for Parabolic Equations AU - Lapinska-Chrzczonowicz , Magdalena AU - Matus , Piotr JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 303 EP - 319 PY - 2008 DA - 2008/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/813.html KW - exact difference scheme, difference scheme with an arbitrary order of accuracy, parabolic equation, system of ordinary differential equations. AB -

The Cauchy problem for the parabolic equation $$\frac{∂u}{∂t} =\frac{∂}{∂x} (k(x,t) \frac{∂u}{∂x}) + f(u,x,,t), x \in R, t > 0,$$ $$u(x,0) = u_0(x), x\in R,$$ is considered. Under conditions $u(x, t) = X(x)T_1(t) + T_2 (t)$, $\frac{∂u}{∂x} \neq 0$, $k(x,t)=k_1(x)k_2(t)$, $f(u,x,t) = f_1(x,t)f_2(u)$, it is shown that the above problem is equivalent to a system of two first-order ordinary differential equations for which exact difference schemes with special Steklov averaging and difference schemes with any order of approximation are constructed on the moving mesh. On the basis of this approach, the exact difference schemes are constructed also for boundary-value problems and multi-dimensional problems. Presented numerical experiments confirm the theoretical results investigated in the paper.

Magdalena Lapinska-Chrzczonowicz & Piotr Matus. (1970). Exact Difference Schemes for Parabolic Equations. International Journal of Numerical Analysis and Modeling. 5 (2). 303-319. doi:
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