A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation
The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation...
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| Format: | Article |
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MDPI AG
2022-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/5/259 |
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| author | Ren Liu Xiaozhong Yang Peng Lyu |
| author_facet | Ren Liu Xiaozhong Yang Peng Lyu |
| author_sort | Ren Liu |
| collection | DOAJ |
| description | The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> under strong regularity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mi>α</mi></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE. |
| first_indexed | 2024-03-10T03:52:35Z |
| format | Article |
| id | doaj.art-cc4ac22369ab4497a368eb10137b8128 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T03:52:35Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-cc4ac22369ab4497a368eb10137b81282023-11-23T11:03:39ZengMDPI AGFractal and Fractional2504-31102022-05-016525910.3390/fractalfract6050259A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher EquationRen Liu0Xiaozhong Yang1Peng Lyu2Institute of Information and Computation, School of Mathematics and Physics, North China Electric Power University, Beijing 102206, ChinaInstitute of Information and Computation, School of Mathematics and Physics, North China Electric Power University, Beijing 102206, ChinaInstitute of Information and Computation, School of Mathematics and Physics, North China Electric Power University, Beijing 102206, ChinaThe fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mrow><mn>2</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> under strong regularity and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>τ</mi><mi>α</mi></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.https://www.mdpi.com/2504-3110/6/5/259inhomogeneous TFFEHASC-N difference schemeunconditional stabilityconvergence ordernumerical tests |
| spellingShingle | Ren Liu Xiaozhong Yang Peng Lyu A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation Fractal and Fractional inhomogeneous TFFE HASC-N difference scheme unconditional stability convergence order numerical tests |
| title | A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation |
| title_full | A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation |
| title_fullStr | A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation |
| title_full_unstemmed | A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation |
| title_short | A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation |
| title_sort | new parallelized computation method of hasc n difference scheme for inhomogeneous time fractional fisher equation |
| topic | inhomogeneous TFFE HASC-N difference scheme unconditional stability convergence order numerical tests |
| url | https://www.mdpi.com/2504-3110/6/5/259 |
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