Abstract
This article examines a compact scheme employing fuzzy transform via exponential basis to solve nonlinear stationary convection–diffusion equations. The scheme executes approximated fuzzy components which estimate the solution values with fourth-order accuracy in an optimal computing time. Such an arrangement associates the approximated fuzzy components with solution values by a linear system. The Jacobian matrices in the scheme are monotone and irreducible. The proof of convergence is briefly discussed. Numerical simulations with nonlinear and linear convection–diffusion equations occurring in quantum mechanics and rheological Carreau fluid will be examined to corroborate the new scheme's utility and efficiency of computational convergence order.
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Funding
Kritika acknowledges the financial support from the University Grants Commission, India, through a research fellowship (NTA Ref. no. 201610085905) to conduct this research work.
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Appendices
Appendix 1: Values of undetermined coefficients with \(\zeta =(p-1)(q-1)\)
\({a}_{i,\mathcal{j}}^{(1)}\) | \({a}_{i,\mathcal{j}}^{(2)}\) | \(\left({p}^{2}+1\right)\left({\beta }^{2}\left({q}^{2}+1\right){h}^{2}-2{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}p{\zeta }^{2}\}\) |
\({a}_{i+1,\mathcal{j}}^{(1)}\) | \({a}_{i-1,\mathcal{j}}^{(2)}\) | \(-\left({\beta }^{2}\left({q}^{2}+1\right){h}^{2}-2{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}{p}^{2}-({p}^{2}-p+1){(p-1)}^{2}\right)/\{{\zeta }^{2}{\alpha }^{2}{\beta }^{2}{h}^{4}p\}\) |
\({a}_{i-1,\mathcal{j}}^{\left(1\right)}\) | \({a}_{i+1,\mathcal{j}}^{\left(2\right)}\) | \(-\left({\beta }^{2}\left({q}^{2}+1\right){h}^{2}-2{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}\pm 1}^{\left(1\right)}\) | \({a}_{i,\mathcal{j}\pm 1}^{\left(2\right)}\) | \(-\left({p}^{2}+1\right)\left({\beta }^{2}{h}^{2}q-{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}p{\zeta }^{2}\}\) |
\({a}_{i+1,\mathcal{j}\pm 1}^{(1)}\) | \({a}_{i-1,\mathcal{j}\pm 1}^{(2)}\) | \(\left({\beta }^{2}q{h}^{2}-{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}{p}^{2}-({p}^{2}-p+1){(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}p{\zeta }^{2}\}\) |
\({a}_{i-1,\mathcal{j}\pm 1}^{(1)}\) | \({a}_{i+1,\mathcal{j}\pm 1}^{(2)}\) | \(\left({\beta }^{2}{h}^{2}q-{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}}^{\left(3\right)}\) | \({a}_{i,\mathcal{j}}^{\left(4\right)}\) | \(\left({q}^{2}+1\right)\left({\beta }^{2}{h}^{2}q-{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}\left({p}^{2}+1\right){h}^{2}-{2\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}q{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}+1}^{(3)}\) | \({a}_{i,\mathcal{j}-1}^{(4)}\) | \(-\left({\alpha }^{2}{h}^{2}\left({p}^{2}+1\right)-2{(p-1)}^{2}\right)\left({\beta }^{2}{h}^{2}{q}^{2}-({q}^{2}-q+1){(q-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}q{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}-1}^{\left(3\right)}\) | \({a}_{i,\mathcal{j}+1}^{\left(4\right)}\) | \(-\left({\beta }^{2}{h}^{2}q-{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}{h}^{2}\left({p}^{2}+1\right)-2{\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i\pm 1,\mathcal{j}+1}^{(3)}\) | \({a}_{i\pm 1,\mathcal{j}-1}^{(4)}\) | \(\left({\beta }^{2}{q}^{2}{h}^{2}-({q}^{2}-q+1){(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}q{\zeta }^{2}\}\) |
\({a}_{i\pm 1,\mathcal{j}}^{(3)}\) | \({a}_{i\pm 1,\mathcal{j}}^{(4)}\) | \(-\left({q}^{2}+1\right)\left({\beta }^{2}{h}^{2}q-{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}q{\zeta }^{2}\}\) |
\({a}_{i\pm 1,\mathcal{j}-1}^{(3)}\) | \({a}_{i\pm 1,\mathcal{j}+1}^{(4)}\) | \(\left({\beta }^{2}{h}^{2}q-{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i\pm 1,\mathcal{j}+1}^{\left(0\right)}\) | \({a}_{i\pm 1,\mathcal{j}-1}^{\left(0\right)}\) | \(\left({\beta }^{2}{h}^{2}q-{(q-1)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i+1,\mathcal{j}}^{\left(0\right)}\) | \({a}_{i-1,\mathcal{j}}^{\left(0\right)}\) | \(-\left({\beta }^{2}\left({q}^{2}+1\right){h}^{2}-2{\left(q-1\right)}^{2}\right)\left({\alpha }^{2}{h}^{2}p-{\left(p-1\right)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}+1}^{\left(0\right)}\) | \({a}_{i,\mathcal{j}-1}^{\left(0\right)}\) | \(-\left({\beta }^{2}{h}^{2}q-{(q-1)}^{2}\right)\left({\alpha }^{2}\left({p}^{2}+1\right){h}^{2}-2{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({a}_{i,\mathcal{j}}^{\left(0\right)}\) | \({a}_{i,\mathcal{j}}^{\left(0\right)}\) | \(\left({\beta }^{2}\left({q}^{2}+1\right){h}^{2}-2{(q-1)}^{2}\right)\left({\alpha }^{2}\left({p}^{2}+1\right){h}^{2}-2{(p-1)}^{2}\right)/\{{\alpha }^{2}{\beta }^{2}{h}^{4}{\zeta }^{2}\}\) |
\({\tau }_{i+1,\mathcal{j}}\) | \({\tau }_{i-1,\mathcal{j}}\) | \(-\{{\alpha }^{2}{h}^{2}p+{(p-1)}^{2}\}/\{{\alpha }^{2}{{h}^{2}\left(p-1\right)}^{2}\}\) |
\({\tau }_{i,\mathcal{j}+1}\) | \({\tau }_{i,\mathcal{j}-1}\) | \(-\{-{\beta }^{2}{h}^{2}q+{(q-1)}^{2}\}/\{{\beta }^{2}{{h}^{2}\left(q-1\right)}^{2}\}\) |
\({\tau }_{i,\mathcal{j}}\) | \({\tau }_{i,\mathcal{j}}\) | \(\{\left({\beta }^{2}{h}^{2}\left({{(pq-1)}^{2}+(p-q)}^{2}\right)-2{\zeta }^{2}\right){\alpha }^{2}-2{\zeta }^{2}{\beta }^{2}\}/\{{\alpha }^{2}{\beta }^{2}{h}^{2}{\zeta }^{2}\}\) |
Appendix 2: Matrices appearing in Eqs. (50) and (51)
\({{\varvec{H}}}_{1}\) | \(\left[\begin{array}{c}\begin{array}{cc}{\alpha }^{2}\left(5{B}_{i,\mathcal{j}}+h\left({A}_{i,\mathcal{j}}+5{B}_{i,\mathcal{j}}^{x}\right)\right)& -{\beta }^{2}\left({C}_{i,\mathcal{j}}+h{C}_{i,\mathcal{j}}^{x}\right)\end{array}\\ \begin{array}{cc}{\alpha }^{2}\left(5{B}_{i,\mathcal{j}}-h\left({A}_{i,\mathcal{j}}+5{B}_{i,\mathcal{j}}^{x}\right)\right)& -{\beta }^{2}\left({C}_{i,\mathcal{j}}-h{C}_{i,\mathcal{j}}^{x}\right)\end{array}\end{array}\right]\) |
\({{\varvec{H}}}_{2}\) | \(\left[\begin{array}{c}\begin{array}{ccc}{A}_{i,\mathcal{j}}+\left({A}_{i,\mathcal{j}}^{x}+2{\alpha }^{2}{B}_{i,\mathcal{j}}\right)h& -h{C}_{i,\mathcal{j}}{\beta }^{2}& {A}_{i,\mathcal{j}}+h{A}_{i,\mathcal{j}}^{x}\end{array}\\ \begin{array}{ccc}{A}_{i,\mathcal{j}}-\left({A}_{i,\mathcal{j}}^{x}+2{\alpha }^{2}{B}_{i,\mathcal{j}}\right)h& h{C}_{i,\mathcal{j}}{\beta }^{2}& {A}_{i,\mathcal{j}}-h{A}_{i,\mathcal{j}}^{x}\end{array}\end{array}\right]\) |
\({{\varvec{H}}}_{3}\) | \(\left[\begin{array}{c}\begin{array}{ccc}-4\left({B}_{i,\mathcal{j}}+h{B}_{i,\mathcal{j}}^{x}\right)& h{A}_{i,\mathcal{j}}& 2\left({C}_{i,\mathcal{j}}+h{C}_{i,\mathcal{j}}^{x}\right)\end{array}\\ \begin{array}{ccc}-4\left({B}_{i,\mathcal{j}}-h{B}_{i,\mathcal{j}}^{x}\right)& -h{A}_{i,\mathcal{j}}& 2\left({C}_{i,\mathcal{j}}-h{C}_{i,\mathcal{j}}^{x}\right)\end{array}\end{array}\right]\) |
\({{\varvec{H}}}_{4}\) | \(\left[\begin{array}{c}\begin{array}{cc}{B}_{i,\mathcal{j}}& -2{C}_{i,\mathcal{j}}\end{array}\\ \begin{array}{cc}{-B}_{i,\mathcal{j}}& 2{C}_{i,\mathcal{j}}\end{array}\end{array}\right]\) |
\({{\varvec{T}}}_{1}\) | \(\left[\begin{array}{c}\begin{array}{cc}{\alpha }^{2}\left({B}_{i,\mathcal{j}}^{y}+h{B}_{i,\mathcal{j}}\right)& -{\beta }^{2}\left(\left({A}_{i,\mathcal{j}}+5{C}_{i,\mathcal{j}}^{y}\right)h+5{C}_{i,\mathcal{j}}\right)\end{array}\\ \begin{array}{cc}{\alpha }^{2}\left({-B}_{i,\mathcal{j}}^{y}+h{B}_{i,\mathcal{j}}\right)& {\beta }^{2}\left(\left({A}_{i,\mathcal{j}}+5{C}_{i,\mathcal{j}}^{y}\right)h-5{C}_{i,\mathcal{j}}\right)\end{array}\end{array}\right]\) |
\({{\varvec{T}}}_{2}\) | \(\left[\begin{array}{c}\begin{array}{ccc}{A}_{i,\mathcal{j}}+h{A}_{i,\mathcal{j}}^{y}& -h{B}_{i,\mathcal{j}}{\alpha }^{2}& {A}_{i,\mathcal{j}}+\left({A}_{i,\mathcal{j}}^{y}+2{\beta }^{2}{C}_{i,\mathcal{j}}\right)h\end{array}\\ \begin{array}{ccc}{A}_{i,\mathcal{j}}-h{A}_{i,\mathcal{j}}^{y}& h{B}_{i,\mathcal{j}}{\alpha }^{2}& {A}_{i,\mathcal{j}}-\left({A}_{i,\mathcal{j}}^{y}+2{\beta }^{2}{C}_{i,\mathcal{j}}\right)h\end{array}\end{array}\right]\) |
\({{\varvec{T}}}_{3}\) | \(\left[\begin{array}{c}\begin{array}{ccc}2\left({B}_{i,\mathcal{j}}+h{B}_{i,\mathcal{j}}^{y}\right)& h{A}_{i,\mathcal{j}}& -4\left({C}_{i,\mathcal{j}}+h{C}_{i,\mathcal{j}}^{y}\right)\end{array}\\ \begin{array}{ccc}2\left({B}_{i,\mathcal{j}}-h{B}_{i,\mathcal{j}}^{y}\right)& -h{A}_{i,\mathcal{j}}& -4\left({C}_{i,\mathcal{j}}-h{C}_{i,\mathcal{j}}^{y}\right)\end{array}\end{array}\right]\) |
\({{\varvec{T}}}_{4}\) | \(\left[\begin{array}{c}\begin{array}{cc}{2B}_{i,\mathcal{j}}& -{C}_{i,\mathcal{j}}\end{array}\\ \begin{array}{cc}{-2B}_{i,\mathcal{j}}& {C}_{i,\mathcal{j}}\end{array}\end{array}\right]\) |
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Jha, N., Kritika Exponential Basis Approximated Fuzzy Components High-Resolution Compact Discretization Technique for 2D Convection–Diffusion Equations. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00616-9
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DOI: https://doi.org/10.1007/s12591-022-00616-9
Keywords
- Fuzzy transform
- Exponential basis
- Schrödinger equation
- Graetz–Nusselt equation
- Convection–diffusion equation
- Convergence analysis