KMUTT FIXED POINT LABORATORY GROUP WORK BOOKLET

KMUTT

CONTENTS 2
3
Contents 5
Seminar Schedule
Abstracts

2

GROUP WORK SCHEDULE

KMUTT Fixed Point Laboratory Group Work Schedule

September 20, 2021

1:30 – 2:30 PM Sakan Termkaew, Urairat Deepan,
Wiparat Worapitpong, & Yutthagan Chummongcol
The Gossip Approach to Optimization Algorithm on CAT(0)
spaces

September 27, 2021

1:30 – 2:30 PM Saqib Murtaza, Talha Anwar, & Asifa
Formulation and Simulation of Brinkman-Type Nanofluid
Model by Using Generalized Fourier’s and Fick’s Laws

2:30 – 3:30 PM Muhammad Arif, Muhammad Ramzan, Dolat Khan
& Muhammad Jabir Khan
Fractional Model of Couple Stress Fluid for Generalized Cou-
ette Flow: A Comparative Analysis of Atangana-Baleanu and
Caputo-Fabrizio Fractional Derivatives

October 4, 2021

1:30 – 2:30 PM J. Abubakar, G. H. Taddele & A. H. Ibrahim
A Mann-Type Iterative Method for Solving Split Variational In-
clusion Problem with Applications

2:30 – 3:30 PM Natthaya Boomyam, Premyuda Dechboon
& Wudthichai Onsod
The basic form of Ekeland variationl principle to an existence
of the estimate point

3

October 11, 2021

1:30 – 2:30 PM Auwalu Hamisu Usman, Adamu Yusuf Inuwa
& Khanitin Muangchoo-in
Existence, Uniqueness and Stability of Fractional Epidemic
Model

2:30 – 3:30 PM Kittisak Amnuaykarn & Pawicha Phairatchatniyom
& Thittaya Vattanavikkit
Best proximity points of generalized α − ψ−Geraghty proximal
contractions in generalized metric spaces

October 18, 2021

1:30 – 2:30 PM Sani Aji, Mahmoud Muhammad Yahaya & Sani Salisu
A Descent Derivative-free Algorithm for System of Nonlinear
Equations and Signal Reconstruction

2:30 – 3:30 PM Musa Ahmed Demba, Yasir Arfat & Arzuka Ibrahim
A Trigonometrically Adapted Embedded Pair of Explicit Runge-
Kutta-Nyström Methods to Solve Periodic SystemsImage In-
painting via the Combination of Sparse Representations and
a Variational Approach

October 25, 2021

1:30 – 2:30 PM Saknarin Channark, Petcharaporn Yodjai
& Kanokwan Kratuloek
An Extend on a Decoupled Method for Image Inpainting with
Patch-based Low Rank Regulariztion

4

ABSTRACTS

Fixed Point Laboratory Group Work Series, 1st Semester, 2021
Fixed Point Theory and Applications Research Group,

Center of Excellence in Theoretical and Computational Science (TaCS-CoE),
Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Thailand

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

The Gossip Approach to Optimization
Algorithm on CAT(0) spaces

[ CAT Gossip ]
Sakan Termkaew & Urairat Deepan & Wiparat Worapitpong &

Yutthagan Chummongcol

email: [email protected]; [email protected];
[email protected]; [email protected]

Abstract
The goal of this work is to propose the concept of a gossip algorithms. This
technique is useful and extensive in the area of network system, especially, in a
consensus situation. Such point can be happened whenever sensor data is estab-
lished in linear spaces. In nonlinear data spaces this algorithms also considered.
We study Random Pairwise Gossip (RPG) algorithm for solving the consensus
problem on geodesic spaces with curvature bounded above by 0. And we consider
the sufficient condition for guarantee the convergence of our algorithm.

Keywords: CAT(0) space, Consensus problem, Gossip optimization

References

[1] Bellachehab, A., and Jérémie J. Random pairwise gossip on CAT (0) metric spaces.
In 53rd IEEE Conference on Decision and Control. 2014:5593-5598. Available from:
doi:10.1109/CDC.2014.7040264.

[2] Bridson, M. R. and Haefliger, A., Metric Spaces of Non-Positive Curvature, Springer-
Verlag, Berlin, 1999: Vol. 319. Available from: doi:10.1007/978-3-662-12494-9

[3] Nedić, A., Pang, J. S., Scutari, G., and Sun, Y. Multi-agent Optimization. Lecture Notes
in Mathematics. 2018. Available from: doi:10.1007/978-3-319-97142-1

6

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

Formulation and Simulation of Brinkman-Type
Nanofluid Model by Using Generalized
Fourier’s and Fick’s Laws

[ Alpha Group ]

Saqib Murtaza & Talha Anwar & Asifa

email: [email protected]; [email protected]; [email protected]

Abstract
Solar energy has received a lot of attention from researchers in the present era. The reasons
appear to be twofold: first, the researchers are interested in developing novel technologies such
as solar collectors and solar water heaters. Secondly, innovative ways to improve the perfor-
mance of solar energy equipment are being used. Keeping in mind the above stated significance
of nanofluids, this article aims to investigate the Caputo time fractional for Brinkman-type
nanofluid in between two parallel vertical plate. The Caputo fractional model has been for-
mulated by utilizing the fractionalized Fourier’s and Fick’s laws. The homogeneous mixture of
different types of ultrafine solid nanoparticles i.e., GO, Al2O3 and T iO2 are uniformly dispersed
in regular water (H2O) Exact solution of the formulated model has been obtained by the joint
use of Fourier sine and the Laplace transform techniques. Closed form solutions are obtained in
terms of Mittag-Leffler function. To interpret the physical arguments of embedded parameters,
various plots are portrayed in graphs and also computed numerically. It is worthy to note that
the heat transfer efficiency of regular water has been improved 25% by using GO nanoparticles,
23.98% by using Al2O3 and 20.88% by using T iO2

Keywords: Generalized Fourier’s and Fick’s law, Modeling and Simulation, Caputo Fractional
Derivative, Mittag-Leffler Function, Brinkman-Type Nanofluid.

References
[1] Ahmad, J., Ali, F., Murtaza, S., & Khan, I. (2021). Caputo Time Fractional Model
Based on Generalized Fourier’s and Fick’s Laws for Jeffrey Nanofluid: Applications in
Automobiles. Mathematical Problems in Engineering, 2021.
[2] Sheikh, N. A., Ching, D. L. C., Khan, I., Sakidin, H. B., Jamil, M., Khalid, H. U.,
& Ahmed, N. (2021). Fractional model for MHD flow of Casson fluid with cadmium
telluride nanoparticles using the generalized Fourier’s law. Scientific reports, 11(1),
1-21.
[3] Sheikh, N. A., Chuan Ching, D. L., Khan, I., & Sakidin, H. B. (2020). Generalization
of the Convective Flow of Brinkman-Type Fluid Using Fourier’s and Fick’s Laws: Exact
Solutions and Entropy Generation. Mathematical Problems in Engineering, 2020.

7

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

Fractional Model of Couple Stress Fluid for
Generalized Couette Flow: A Comparative

Analysis of AtanganaBaleanu and
CaputoFabrizio Fractional Derivatives

[ Pak Maths ]

Muhammad Arif, Muhammad Ramzan, Dolat Khan & Muhammad Jabir Khan

[email protected]; [email protected]; [email protected];
[email protected]

Abstract
Different from a Newtonian fluid, Couple stress fluid (CSF), includes a new material constant
which is responsible for couple stress and the lubricant viscosity. This material constant comes
with the 4th order spatial derivative term, and due to this higher order derivative term in the
momentum equation, this fluid (CSF) is comparatively less investigated even for the classical
fluid problems. This article aims to study the fractional model of CSF, based on AB fractional
derivatives definition. Since this AB definition is new, therefore, for the sake of comparison
and correctness, this problem is also solved using (CF) fractional derivative definition. CSF is
considered to flow between two parallel plates one of which is at rest and the other is moving
with constant velocity in the presence of external pressure. This type of flow situations is usu-
ally called as generalized Couette flow. The problem is solved for the exact solution using the
Laplace Fourier transforms. CSF velocity obtained via AB fractional derivative is compared
with CSF velocity obtained via CF fractional derivative approach and the results obtained are
shown graphically. In limiting sense, the present CSF solutions are reduced to a similar New-
tonian fluid problem solution in the absence of external pressure gradient.

Keywords: Couple stress fluid, AtanganaBaleanu and CaputoFabrizio .

References
[1] M. Arif, F. Ali, I. Khan and K. S. Nisar, "A Time Fractional Model With Non-Singular
Kernel the Generalized Couette Flow of Couple Stress Nanofluid," in IEEE Access, vol.
8, pp. 77378-77395, 2020, doi: 10.1109/ACCESS.2020.2982028.
[2] Arif, M., Kumam, P., Kumam, W., Khan, I., Ramzan, M. (2021). A Fractional Model
of Casson Fluid with Ramped Wall Temperature: Engineering Applications of Engine
Oil. Computational and Mathematical Methods, e1162.

8

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

A Mann-Type Iterative Method for Solving Split
Variational Inclusion Problem with Applications

[ Jamil, Guash and Abdul ]

J. Abubakar & G. H. Taddele & A. H. Ibrahim

email: [email protected]; [email protected] ;
[email protected]

Abstract
A new strong convergence iterative method for solving a split variational inclusion problem
involving a bounded linear operator and two maximally monotone mappings is proposed in
this article. The study considers an iterative scheme comprised of inertial extrapolation step
together with the Mann-type step. A strong convergence theorem of the iterates generated by
the proposed iterative scheme is given under suitable conditions. In addition, methods for solv-
ing variational inequality problems and split convex feasibility problems are derived from the
proposed method. Applications of solving Nash-equilibrium problems and image restoration
problems are solved using the derived methods to demonstrate the implementation of the pro-
posed methods. Numerical comparisons with some existing iterative methods are also presented.

Keywords: Hadamard manifolds

References
[1] Lions PL, Mercier B. Splitting algorithms for the sum of two nonlinear operators.
SIAM J. Numer Anal. 1979;16(6):964–979. Available from: https://doi.org/10.1137/
0716071.
[2] Takahashi W, Wong NC & Yao JC. Two generalized strong convergence theorems of
Halpern’s type in Hilbert spaces and applications. Taiwanese J Math. 2012;16(3):1151–
1172. Available from: https://doi.org/10.11650/twjm/1500406684.
[3] Byrne C., Censor Y., Gibali A. and Reich S., The split common null point problem, J.
Nonlinear Convex Anal, 2012;13, 759–775.
[4] Cholamjiak P., Thong D. V. and Cho Y. J., A novel inertial projection and contraction
method for solving pseudomonotone variational inequality problems, Acta Applicandae
Mathematicae, 2020;169, 217–245.

9

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

The basic form of Ekeland variationl principle to
an existence of the estimate point

[ Fixed Point members ]
Natthaya Boomyam & Premyuda Dechboon & Wudthichai Onsod
email: [email protected]; [email protected];

[email protected]
Abstract
In this talk, we would like to explain Ekeland variationl principle which is one of the most
important results from nonlinear analysis. Since it can be used to establish the existence of the
estimate optimal and equilibrium, therefore, we finally illustrate extension versions of Ekeland
variationl principle in this talk.
Keywords: Ekeland variationl principle, Equilibrium, Nonlinear analysis, Optimal point

References
[1] Al-Homidan S., Ansari Q.H. & Kassay G. Vectorial form of Ekeland variational principle
with applications to vector equilibrium problems. Optimization. 2020;69(3):415–436.
Available from: https://doi.org/10.1080/02331934.2019.1589469.
[2] Sitthithakerngkiet K. & Plubtieng S. Vectorial form of Ekeland-type variational principle.
Fixed Point Theory Appl. 2012;127, 11. Available from: https://doi.org/10.1186/
1687-1812-2012-127.

10

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

Existence, Uniqueness and Stability of
Fractional Epidemic Model

[ Epidemic model Group ]
Auwalu Hamisu Usman & Adamu Yusuf Inuwa & Khanitin Muangchoo-in
email: [email protected]; [email protected]; [email protected]
Abstract
This paper describes the existence and stability of the epidemic model with fractional-order
derivative (FOD) in Atangana-Baleanu (AB) sense. Some new results are handled by using
the Sumudu transform. The existence as well as uniqueness of the equilibrium solution are
presented using Banach fixed point theorem. Moreover, sensitivity analysis complemented by
simulations are performed to determine how changes in parameters affect the dynamical behav-
ior of the system. The numerical simulations are carried out using a predictor-corrector scheme
to demonstrate the obtained results.

Keywords: Mathematical Model, Atangana-Baleanu derivative, Sumudu Transform, Exis-
tence and uniqueness, Numerical simulation.

References
[1] Caputo M. & Fabrizio M. A new definition of fractional derivative with out singular
kernel, Progr. Fract. Differ. Appl. 2015;(1):73–85.
[2] El-Saka H.A.A. The fractional-order SIS epidemic model with variable population size,
J.Egyptian. Math. Soci. 2015;(22): 50–54
[3] Pang J, Cui JA & Zhou X. Dynamical behavior of a hepatitis B virus transmission model
with vaccination. J Theor Biol 2010;(265):572–578.
[4] Sharomi O. & Gumel A. B. Curtailing smoking dynamics: a mathematical modeling
approach Applied Mathematics and Computation. 2008; 2(195): 475–499.

11

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

Best proximity points of generalized
α − ψ−Geraghty proximal contractions in

generalized metric spaces

[ Best proximity points of MheeWiNeen ]
Kittisak Amnuaykarn & Pawicha Phairatchatniyom & Thittaya Vattanavikkit

email: [email protected]; [email protected];
[email protected]

Abstract
In this article, we introduce the new class of α−ψ−Geraghty proximal contractions called “gen-
eralized α − ψ−Geraghty proximal contractions” in generalized metric spaces. Furthermore, we
set up some best proximity point results for proposed mapping. Our results improve and extend
several related results in the literature.
Keywords: Fixed point theory, Contraction mapping, Best proximity point, Geraghty proxi-
mal contraction, Generalized metric space.

References
[1] Asadi M, Karapnar E and Kumar A. α − ψ−Geraghty contractions on generalized metric
spaces. Journal of Inequalities and Applications. 2014; 423.
[2] Kumssa LB. Best proximity point of modified Suzuki-Edelstein-Geraghty type proximal
contractions. Engineering and Applied Science Letters. 2020; 3(4): 94–104.

12

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

A Descent Derivative-free Algorithm for System
of Nonlinear Equations and Signal
Reconstruction

[ SMS ]

Sani Aji & Mahmoud Muhammad Yahaya & Sani Salisu

email: [email protected]; [email protected]; [email protected]

Abstract
The main purpose of this work is to propose a descent derivative free algorithm for solving sys-
tem of nonlinear equations and show its application in reconstructing a distorted signal. The
proposed algorithm is descent and does not require computation of any gradient. The global
convergence of the proposed algorithm is proved under the assumptions that the mapping under
consideration is Lipschitz continuous and monotone. To depict the efficiency of the algorithm,
numerical experiments are given in comparison with some existing ones. The numerical exper-
iments have shown that the new method has some advantages in solving nonlinear equations
and handling signal restoration problems

Keywords: Nonlinear equations, Signal recovery, Conjugate gradient methed

References
[1] W. La Cruz, J. M. Martínez, and M. Raydan, Spectral residual method without gradient
information for solving large-scale nonlinear systems of equations, Math. Comput., vol.
75, no. 255, pp. 1429-1448, 2006.
[2] E. D. Dolan, J. J. More, Benchmarking optimization software with performance profiles.
ť Math. Program., 91 (2002), 201-213
[3] J. K. Liu, S. J. Li, Multivariate spectral DY-type projection method for convex con-
strained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.
[4] Y. H. Xiao, Q. Y. Wang, Q. J. Hu, Non-smooth equations based method for 1-norm
problems with applications to compressed sensing, Nonlinear Anal.: Theory, 74 (2011),
3570-3577.
[5] M. V. Solodov, B. F. Svaiter, A globally convergent inexact newton method for systems
of monotone equations, In: Reformulation: Nonsmooth, piecewise smooth, semismooth
and smoothing methods, 1998, 355369.

13

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

A Trigonometrically Adapted Embedded Pair
of Explicit Runge-Kutta-Nyström Methods

to Solve Periodic Systems

[ Shining Star ]
Musa Ahmed Demba & Yasir Arfat & Arzuka Ibrahim

email: [email protected]; [email protected]; [email protected]

Abstract
In this work a 5(3) pair of explicit trigonometrically adapted Runge-Kutta-Nyström methods
with four stages is derived based on an explicit pair appeared in the literature. The new adapted
method is able to integrate exactly the usual test equation: y′′ = −w2y. The local trunca-
tion error of the new method is obtained, proving that the algebraic order of convergence is
maintained. The stability interval of the new method is obtained, showing that the proposed
method is absolutely stable. The numerical experiments performed demonstrate the robustness
of the new embedded pair in comparison with some standard codes available in the literature.

Keywords: Trigonometrically-fitted method, Runge-Kutta-Nyström, Periodic Problems, Ini-
tial Value Problems

References
[1] H. Van de Vyver, "A Runge-Kutta-Nyström pair for the numerical integration of per-
turbed oscillators", Computer Physics Communications, 2005;167:129-142, Available
from: https://doi.org/10.1016/j.cpc.2004.12.011.
[2] H. Van de Vyver, "An embedded exponentially fitted Runge-Kutta-Nyström method
for the numerical solution of orbital problems", New Astronomy, 2006; 11(8): 577-587,
Available from: https://doi.org/10.1016/j.newast.2006.03.001.
[3] H. Van de Vyver, "A 5 (3) pair of explicit Runge-Kutta-Nyström methods for oscillatory
problems", Mathematical and computer modelling, 2007;45(5-6):708-716, Available from:
https://doi.org/10.1016/j.mcm.2006.07.016.

14

1st Semester, 2021
Fixed Point Laboratory Group Work Series,
Fixed Point Theory and Applications Research Group,
Center of Excellence in Theoretical and Computational
Science (TaCS-CoE), Faculty of Science, King
Mongkut’s University of Technology Thonburi

(KMUTT), Thailand

An Extend on a Decoupled Method for Image
Inpainting with Patch-based Low Rank
Regulariztion

[ SPK Denoise ]
Saknarin Channark & Petcharaporn Yodjai & Kanokwan Kratuloek

email: [email protected]; [email protected];
[email protected]

Abstract
We present a decoupled variational method for image inpainting in the image domain in this
paper. The original image inpainting problem is decoupled into three separate minimization
problems with various energy functionals. The first is image denoising using the patch-based
weighted nuclear norm minimization approach, which is a low-rank regularization method
(PWNNM). Second, the gradient of u and its contour e, as well as the contour with a broad
panel of regularization terms, are both flexible.The last is a linear combination in the image do-
main. An iterative algorithm is then obtained by minimizing the three problems alternatingly.
We simplify the denoising procedure by breaking it down into three steps: image decomposition,
patch matrix denoising, and image reconstruction. The numerical algorithm’s convergence is
demonstrated under certain assumptions. The proposed approaches’ efficiency is demonstrated
by numerical experiments and comparisons on various images.

Keywords: Image inpainting, Weighted nuclear norm, Mumford - Shah

References
[1] Li F, Lv X. A Decoupled method for image inpainting with patch-based low rank reg-
ulariztion. Applied Mathematics and Computation. 2017;314:334348. Available from:
https://www.sciencedirect.com/science/article/pii/S009630031730440X.
[2] Foare M, Pustelnik N, Condat L. Semi-Linearized Proximal Alternating Minimization
for a Discrete Mumford - Shah Model. IEEE Transactions on Image Processing. 2020;
29:2176-2189.

15