Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives
The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi&g...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-02-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/6/3/132 |
| _version_ | 1797471542022504448 |
|---|---|
| author | Fan Zhang Hai-Yang Xu Heng-You Lan |
| author_facet | Fan Zhang Hai-Yang Xu Heng-You Lan |
| author_sort | Fan Zhang |
| collection | DOAJ |
| description | The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>H</mi></mrow></semantics></math></inline-formula>-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>H</mi></mrow></semantics></math></inline-formula>-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-approximate solution of the coupled system. Finally, we propose some work for future research. |
| first_indexed | 2024-03-09T19:49:39Z |
| format | Article |
| id | doaj.art-d6bf7935057948169d5ed952666026ff |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T19:49:39Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-d6bf7935057948169d5ed952666026ff2023-11-24T01:14:10ZengMDPI AGFractal and Fractional2504-31102022-02-016313210.3390/fractalfract6030132Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type DerivativesFan Zhang0Hai-Yang Xu1Heng-You Lan2College of Mathematics and Statistics, Sichuan University of Science & Engineering, Zigong 643000, ChinaCollege of Mathematics and Statistics, Sichuan University of Science & Engineering, Zigong 643000, ChinaCollege of Mathematics and Statistics, Sichuan University of Science & Engineering, Zigong 643000, ChinaThe purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>H</mi></mrow></semantics></math></inline-formula>-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mi>H</mi></mrow></semantics></math></inline-formula>-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ε</mi></semantics></math></inline-formula>-approximate solution of the coupled system. Finally, we propose some work for future research.https://www.mdpi.com/2504-3110/6/3/132existence and uniquenesscontinuous dependence and ε-approximationcoupled system of fuzzy fractional partial differential equationsBanach fixed point theoremGronwall inequality of the vector form |
| spellingShingle | Fan Zhang Hai-Yang Xu Heng-You Lan Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives Fractal and Fractional existence and uniqueness continuous dependence and ε-approximation coupled system of fuzzy fractional partial differential equations Banach fixed point theorem Gronwall inequality of the vector form |
| title | Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives |
| title_full | Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives |
| title_fullStr | Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives |
| title_full_unstemmed | Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives |
| title_short | Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo <i>gH</i>-Type Derivatives |
| title_sort | initial value problems of fuzzy fractional coupled partial differential equations with caputo i gh i type derivatives |
| topic | existence and uniqueness continuous dependence and ε-approximation coupled system of fuzzy fractional partial differential equations Banach fixed point theorem Gronwall inequality of the vector form |
| url | https://www.mdpi.com/2504-3110/6/3/132 |
| work_keys_str_mv | AT fanzhang initialvalueproblemsoffuzzyfractionalcoupledpartialdifferentialequationswithcaputoighitypederivatives AT haiyangxu initialvalueproblemsoffuzzyfractionalcoupledpartialdifferentialequationswithcaputoighitypederivatives AT hengyoulan initialvalueproblemsoffuzzyfractionalcoupledpartialdifferentialequationswithcaputoighitypederivatives |