Published December 1, 2021
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On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative
- 1. Imam Mohammad ibn Saud Islamic University
- 2. Hodeidah University
- 3. Imam Abdulrahman Bin Faisal University
Description
Abstract In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder's and Banach's fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.
Translated Descriptions
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This is an automatic machine translation with an accuracy of 90-95%
Translated Description (Arabic)
نبذة مختصرة في هذه الورقة، ندرس فئة من مسائل القيمة الأولية لمعادلة تفاضلية كسرية ضمنية غير خطية مع ظروف غير محلية تتضمن المشتق الكسري Atangana - Baleanu - Caputo. يعتمد المشغل الكسري المطبق على نواة غير مفردة وغير محلية. ثم نستمد صيغة للحل من خلال معادلات التكامل الوظيفي الكسرية المكافئة للمسألة المقترحة. يتم الحصول على الوجود والتفرد عنطريق نظريات النقاط الثابتة لشودر وبنك. علاوة على ذلك، تتم مناقشة نوعين من الاعتماد المستمر لحلول مثل هذه المعادلات. وأخيرًا، تتضمن الورقة مثالين لإثبات صحة النتائج الرئيسية.Translated Description (English)
Abstract In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means ofSchauder's and Banach's fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.Translated Description (French)
Résumé Dans cet article, nous étudions une classe de problèmes de valeur initiale pour une équation différentielle fractionnaire implicite non linéaire avec des conditions non locales impliquant la dérivée fractionnaire Atangana-Baleanu-Caputo. L'opérateur fractionnaire appliqué est basé sur un noyau non singulier et non local. Ensuite, nous dérivons une formule pour la solution à travers les équations intégrales fonctionnelles fractionnaires équivalentes au problème proposé. L'existence et l'unicité sont obtenues au moyen des théorèmes à virgule fixe de Schauder et de Banach. De plus, deux types de dépendance continue de solutions à de telles équations sont discutés. Enfin, le document comprend deux exemples pour étayer la validité des principaux résultats.Translated Description (Spanish)
Resumen En este artículo, estudiamos una clase de problemas de valor inicial para una ecuación diferencial fraccional implícita no lineal con condiciones no locales que involucran la derivada fraccional de Atangana-Baleanu-Caputo. El operador fraccionario aplicado se basa en un núcleo no singular y no local. Luego derivamos una fórmula para la solución a través de las ecuaciones integrales funcionales fraccionarias equivalentes al problema propuesto. La existencia y singularidad se obtienen por medio de los teoremas de punto fijo de Schauder y Banach. Además, se discuten dos tipos de dependencia continua de soluciones a tales ecuaciones. Finalmente, el documento incluye dos ejemplos para corroborar la validez de los principales resultados.Files
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Additional details
Additional titles
- Translated title (Arabic)
- على مشكلة ضمنية غير محلية تحت المشتق الكسري Atangana - Baleanu - Caputo
- Translated title (English)
- On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative
- Translated title (French)
- Sur un problème implicite non local sous la dérivée fractionnaire Atangana-Baleanu-Caputo
- Translated title (Spanish)
- Sobre un problema implícito no local bajo la derivada fraccionaria de Atangana-Baleanu-Caputo
Identifiers
- Other
- https://openalex.org/W4200076685
- DOI
- 10.1186/s13661-021-01579-6
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