On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition

Mtema James Chin

doi:doi:10.11648/j.ajam.20251305.12

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On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition

Mtema James Chin^*

Published in American Journal of Applied Mathematics (Volume 13, Issue 5)

Received: 5 July 2025 Accepted: 11 August 2025 Published: 12 September 2025

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Abstract

In this paper, we investigate the Existence and Uniqueness of solutions of non-linear fractional boundary value differential equations with integral boundary condition using the method of Upper and Lower solutions. We employed the contraction mapping principle and Schauder fixed point theorems. We find out from the analysis that the solution of the boundary value fractional differential equation exists and is unique. An Adomian decomposition method is also used to construct the algorithm for the numerical solution of the nonlinear fractional differential equation. Further, for the implementation of the Adomian decomposition method, several numerical examples are constructed to demonstrate the applicability, accuracy, efficiency, and effectiveness of the method. The results show that the method is accurate and efficient in approximating the exact solution.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 5)
DOI	10.11648/j.ajam.20251305.12
Page(s)	320-338
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Existence and Uniqueness, Hadamard Fractional Derivatives, Analytic Solutions, Integral Boundary Condition, Banach Space, Numerical Solution

References

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APA Style

Chin, M. J. (2025). On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. American Journal of Applied Mathematics, 13(5), 320-338. https://doi.org/10.11648/j.ajam.20251305.12

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ACS Style

Chin, M. J. On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. Am. J. Appl. Math. 2025, 13(5), 320-338. doi: 10.11648/j.ajam.20251305.12

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AMA Style

Chin MJ. On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. Am J Appl Math. 2025;13(5):320-338. doi: 10.11648/j.ajam.20251305.12

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@article{10.11648/j.ajam.20251305.12,
  author = {Mtema James Chin},
  title = {On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {5},
  pages = {320-338},
  doi = {10.11648/j.ajam.20251305.12},
  url = {https://doi.org/10.11648/j.ajam.20251305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251305.12},
  abstract = {In this paper, we investigate the Existence and Uniqueness of solutions of non-linear fractional boundary value differential equations with integral boundary condition using the method of Upper and Lower solutions. We employed the contraction mapping principle and Schauder fixed point theorems. We find out from the analysis that the solution of the boundary value fractional differential equation exists and is unique. An Adomian decomposition method is also used to construct the algorithm for the numerical solution of the nonlinear fractional differential equation. Further, for the implementation of the Adomian decomposition method, several numerical examples are constructed to demonstrate the applicability, accuracy, efficiency, and effectiveness of the method. The results show that the method is accurate and efficient in approximating the exact solution.
},
 year = {2025}
}

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TY  - JOUR
T1  - On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition

AU  - Mtema James Chin
Y1  - 2025/09/12
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251305.12
DO  - 10.11648/j.ajam.20251305.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 320
EP  - 338
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251305.12
AB  - In this paper, we investigate the Existence and Uniqueness of solutions of non-linear fractional boundary value differential equations with integral boundary condition using the method of Upper and Lower solutions. We employed the contraction mapping principle and Schauder fixed point theorems. We find out from the analysis that the solution of the boundary value fractional differential equation exists and is unique. An Adomian decomposition method is also used to construct the algorithm for the numerical solution of the nonlinear fractional differential equation. Further, for the implementation of the Adomian decomposition method, several numerical examples are constructed to demonstrate the applicability, accuracy, efficiency, and effectiveness of the method. The results show that the method is accurate and efficient in approximating the exact solution.

VL  - 13
IS  - 5
ER  -

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Mtema James Chin

Department of Mathematics, College of Physical Sciences, Federal University of Agriculture, Makurdi, Nigeria

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APA Style

Chin, M. J. (2025). On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. American Journal of Applied Mathematics, 13(5), 320-338. https://doi.org/10.11648/j.ajam.20251305.12

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ACS Style

Chin, M. J. On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. Am. J. Appl. Math. 2025, 13(5), 320-338. doi: 10.11648/j.ajam.20251305.12

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AMA Style

Chin MJ. On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition. Am J Appl Math. 2025;13(5):320-338. doi: 10.11648/j.ajam.20251305.12

Copy | Download

@article{10.11648/j.ajam.20251305.12,
  author = {Mtema James Chin},
  title = {On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition
},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {5},
  pages = {320-338},
  doi = {10.11648/j.ajam.20251305.12},
  url = {https://doi.org/10.11648/j.ajam.20251305.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251305.12},
  abstract = {In this paper, we investigate the Existence and Uniqueness of solutions of non-linear fractional boundary value differential equations with integral boundary condition using the method of Upper and Lower solutions. We employed the contraction mapping principle and Schauder fixed point theorems. We find out from the analysis that the solution of the boundary value fractional differential equation exists and is unique. An Adomian decomposition method is also used to construct the algorithm for the numerical solution of the nonlinear fractional differential equation. Further, for the implementation of the Adomian decomposition method, several numerical examples are constructed to demonstrate the applicability, accuracy, efficiency, and effectiveness of the method. The results show that the method is accurate and efficient in approximating the exact solution.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - On the Existence and Uniqueness of Solution of Non-linear Fractional Differential Equations with Integral Boundary Condition

AU  - Mtema James Chin
Y1  - 2025/09/12
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251305.12
DO  - 10.11648/j.ajam.20251305.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 320
EP  - 338
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251305.12
AB  - In this paper, we investigate the Existence and Uniqueness of solutions of non-linear fractional boundary value differential equations with integral boundary condition using the method of Upper and Lower solutions. We employed the contraction mapping principle and Schauder fixed point theorems. We find out from the analysis that the solution of the boundary value fractional differential equation exists and is unique. An Adomian decomposition method is also used to construct the algorithm for the numerical solution of the nonlinear fractional differential equation. Further, for the implementation of the Adomian decomposition method, several numerical examples are constructed to demonstrate the applicability, accuracy, efficiency, and effectiveness of the method. The results show that the method is accurate and efficient in approximating the exact solution.

VL  - 13
IS  - 5
ER  -

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