Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations
In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-06-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/15/7/1338 |
| _version_ | 1797587425344618496 |
|---|---|
| author | Hasib Khan Jehad Alzabut Wafa F. Alfwzan Haseena Gulzar |
| author_facet | Hasib Khan Jehad Alzabut Wafa F. Alfwzan Haseena Gulzar |
| author_sort | Hasib Khan |
| collection | DOAJ |
| description | In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></semantics></math></inline-formula> in two subclasses as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>1</mn></msub><mo>=</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>2</mn></msub><mo>=</mo><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>1</mn></msub></semantics></math></inline-formula>, the classical derivative is considered for the study of the leukemia growth while in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>2</mn></msub></semantics></math></inline-formula> we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots. |
| first_indexed | 2024-03-11T00:35:54Z |
| format | Article |
| id | doaj.art-1c6ab54ca6f64250bd0223a186a49b58 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T00:35:54Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1c6ab54ca6f64250bd0223a186a49b582023-11-18T21:33:41ZengMDPI AGSymmetry2073-89942023-06-01157133810.3390/sym15071338Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical SimulationsHasib Khan0Jehad Alzabut1Wafa F. Alfwzan2Haseena Gulzar3Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi ArabiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Biotechnology, Shaheed Benazir Bhutto Uniersity, Sheringal, Dir Upper 18000, PakistanIn this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></semantics></math></inline-formula> in two subclasses as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>1</mn></msub><mo>=</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>2</mn></msub><mo>=</mo><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>1</mn></msub></semantics></math></inline-formula>, the classical derivative is considered for the study of the leukemia growth while in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>2</mn></msub></semantics></math></inline-formula> we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots.https://www.mdpi.com/2073-8994/15/7/1338piecewise modified ABC derivativeleukemia mathematical modelexistence of solutionsHyers–Ulam stabilitycrossover behavior |
| spellingShingle | Hasib Khan Jehad Alzabut Wafa F. Alfwzan Haseena Gulzar Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations Symmetry piecewise modified ABC derivative leukemia mathematical model existence of solutions Hyers–Ulam stability crossover behavior |
| title | Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations |
| title_full | Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations |
| title_fullStr | Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations |
| title_full_unstemmed | Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations |
| title_short | Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations |
| title_sort | nonlinear dynamics of a piecewise modified abc fractional order leukemia model with symmetric numerical simulations |
| topic | piecewise modified ABC derivative leukemia mathematical model existence of solutions Hyers–Ulam stability crossover behavior |
| url | https://www.mdpi.com/2073-8994/15/7/1338 |
| work_keys_str_mv | AT hasibkhan nonlineardynamicsofapiecewisemodifiedabcfractionalorderleukemiamodelwithsymmetricnumericalsimulations AT jehadalzabut nonlineardynamicsofapiecewisemodifiedabcfractionalorderleukemiamodelwithsymmetricnumericalsimulations AT wafafalfwzan nonlineardynamicsofapiecewisemodifiedabcfractionalorderleukemiamodelwithsymmetricnumericalsimulations AT haseenagulzar nonlineardynamicsofapiecewisemodifiedabcfractionalorderleukemiamodelwithsymmetricnumericalsimulations |