Browsing by Author "Selvitopi, Harun"

Now showing 1 - 19 of 19

(1+2) Boyutlu Zamana Bağlı Konvektif Katsayılı Sönümlemeli Dalga Tür Konveksiyon-Difüzyon-Reaksiyon Denkleminin Sonlu Farklar Sonlu Elemanlar Yöntemi ile Çözümü
(2024) Adıgüzel, Zeynep; Selvitopi, Harun
Yapılan çalışmaları incelediğimizde görüyoruz ki, birçok mühendislik ve fizik problemleri konveksiyon-difüzyon-reaksiyon denklemi ile matematiksel olarak modellenmektedir. Bu tez çalışmasında ise literatüre ilk kez girecek ve fiziksel problemlere oldukça katkı sağlayacak, hareketli alanlardaki salınımları kontrol etmede büyük role sahip olan iki boyutlu sönümlemeli dalga tür konveksiyon-difüzyon-reaksiyon denkleminin sonlu fark/sonlu elemanlar yöntemi ile çözümü ele alınmıştır. Fortran programlama dili kullanılarak elde edilen sayısal çözümlerin sonuçları zamana bağlı konvektif katsayılı denklemler için farklı zaman adımlarında elde edilmiş, incelenmiş ve yorumlanmıştır.
2 ve 3 Boyutlu Uzayda Zaman Bağımlı Konveksiyon Difüzyon Reaksiyon Denklemlerinin Fizik Bilgili Sinir Ağı ile Çözümü
(2025) Beyazlı, Furkan; Selvitopi, Harun
Birçok mühendislik ve fizik problemi, konveksiyon-difüzyon-reaksiyon (KDR) denklemleri ile matematiksel olarak modellenebilmektedir. Bu tez çalışmasında, ızgarasız bir yöntem olan Fizik Bilgili Sinir Ağı (FBSA) kullanılarak, bir, iki ve üç boyutlu doğrusal olmayan KDR denklemlerinin çözümü gerçekleştirilmiştir. Elde edilen sayısal çözümler, analitik çözümlerle karşılaştırılmış ve L2 hata normu, KOKH ve maksimum mutlak hata gibi metrikler kullanılarak değerlendirilmiştir. Sonuçlar, FBSA yönteminin özellikle yüksek boyutlu ve konveksiyon-baskın problemlerde etkin ve doğru çözümler üretebildiğini göstermektedir. Bu çalışma, literatürde FBSA tabanlı 3D KDR çözümü sunması açısından özgün bir katkı sağlamaktadır.
Anthracene-Pyridine Derivatives as Fluorescent Probes: The Role of Nitrogen Positioning in Bioimaging Performance
(Elsevier Science Sa, 2026) Aydemir, Murat; Haykir, Gulcin; Demir, Seda; Aydemir, Esra; Arslan, Elif; Selvitopi, Harun; Turksoy, Figen
The strategic design of small-molecule fluorescent probes is critical for advancing precision bioimaging in both cellular and microbiological contexts. In this study, we report a series of anthracene-pyridine derivatives-compounds 3a, 3b, and 3c-with nitrogen atoms positioned at the ortho, meta, and para positions, respectively, to investigate how atomic-level substitution patterns govern photophysical properties and imaging performance. Spectroscopic characterization revealed that the para-substituted compound 3c exhibits enhanced intramolecular charge transfer (ICT) character, leading to lower fluorescence quantum yield in polar biological environments due to increased non-radiative decay. In contrast, the meta-substituted derivative 3b maintains a more locally excited (LE)-like emission, producing intense blue fluorescence and demonstrating high selectivity for bacterial imaging, likely due to favourable interactions with nucleic acids or membranes. Ortho-substituted compound 3a also displayed appreciable fluorescence in mammalian cells, though with lower intensity and reduced bacterial uptake. Complementary molecular dynamics simulations revealed that nitrogen positioning influences molecular geometry, conformational stability, and interaction propensities with biological targets, thereby shaping the observed bioimaging performance. Collectively, these findings establish clear structure-property-function relationships, underscoring how fine-tuning nitrogen placement can optimize excited-state behaviour, cellular uptake, and emission output. These insights provide a valuable framework for the rational design of next-generation fluorophores tailored for multicolour and environment-sensitive imaging applications.
Computational Identification of Potential Antiviral Agents Against Monkeypox Virus
(2025) Aydemir, Dr. Murat; Hacimuftuoglu, Ahmet; Arslan, Mehmet Enes; Turkez, Hasan; Akbaba, Yusuf; Selvitopi, Harun; Selvitopi, Zulkuf
This study introduces a drug repurposing approach for monkeypox virus using molecular docking and molecular dynamics (MD) simulations to screen clinically approved drugs for other diseases. Potential candidates were analyzed with AutoDock software in two categories: (1) DNA polymerase inhibitors and (2) RNA polymerase inhibitors. Docking analysis identified seven promising compounds—five DNA polymerase and two RNA polymerase inhibitors—showing strong affinity toward monkeypox targets. MD simulations further confirmed their binding stability. Among them, etravirine (−8.04 kcal/mol) and valaciclovir (−7.57 kcal/mol) exhibited the highest affinity to DNA polymerase active sites at residues ASP347 and ASP462. Moreover, emetine dihydrochloride (−7.17 kcal/mol) demonstrated the strongest binding to the catalytic site of RNA polymerase. Consequently, etravirine and emetine dihydrochloride are proposed as potential therapeutics against monkeypox. Since these compounds are already approved by the U.S. Food and Drug Administration (FDA) for other viral infections, extensive safety testing and high development costs can be avoided, making this repurposing strategy an efficient and cost-effective option for managing monkeypox infection.
Costunolide and Parthenolide Ameliorate MPP Plus Induced Apoptosis in the Cellular Parkinson's Disease Model
(MDPI, 2023) Arslan, Mehmet Enes; Turkez, Hasan; Sevim, Yasemin; Selvitopi, Harun; Kadi, Abdurrahim; Oner, Sena; Mardinoglu, Adil
Monoamine oxidase B (MAO-B) is an enzyme that metabolizes several chemicals, including dopamine. MAO-B inhibitors are used in the treatment of Parkinson's Disease (PD), and the inhibition of this enzyme reduces dopamine turnover and oxidative stress. The absence of dopamine results in PD pathogenesis originating from decreased Acetylcholinesterase (AChE) activity and elevated oxidative stress. Here, we performed a molecular docking analysis for the potential use of costunolide and parthenolide terpenoids as potential MAO-B inhibitors in the treatment of PD. Neuroprotective properties of plant-originated costunolide and parthenolide terpenoids were investigated in a cellular PD model that was developed by using MPP+ toxicity. We investigated neuroprotection mechanisms through the analysis of oxidative stress parameters, acetylcholinesterase activity and apoptotic cell death ratios. Our results showed that 100 mu g/mL and 50 mu g/mL of costunolide, and 50 mu g/mL of parthenolide applied to the cellular disease model ameliorated the cytotoxicity caused by MPP+ exposure. We found that acetylcholinesterase activity assays exhibited that terpenoids could ameliorate and restore the enzyme activity as in negative control levels. The oxidative stress parameter analyses revealed that terpenoid application could enhance antioxidant levels and decrease oxidative stress in the cultures. In conclusion, we reported that these two terpenoid molecules could be used in the development of efficient treatment strategies for PD patients.
Crank-Nicolson/Finite Element Approximation for the Schrödinger Equation in the De Sitter Spacetime
(IOP Publishing Ltd, 2021) Selvitopi, Harun; Zaky, Mahmoud A.; Hendy, Ahmed S.
Central to much of science, engineering, and society today is the building of mathematical models to represent complex processes. Recently, the non-relativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime has been used to derive Schrodinger equations with weighted nonlinear terms In this paper, numerical simulations are constructed to clarify the behavior of the solution in both one- and two-dimensions. These simulations are constructed based on the Crank-Nicolson scheme in the time direction and the Galerkin finite element in the spatial direction. The nonlinear system of algebraic equations resulting from the constructed scheme is solved using Newton's method. It is also demonstrated that the numerical approximation converges to the exact one.
Drug Synergism of Anticancer Action in Combination with Favipiravir and Paclitaxel on Neuroblastoma Cells
(MDPI, 2024) Turkez, Hasan; Arslan, Mehmet Enes; Selvitopi, Harun; Kadi, Abdurrahim; Oner, Sena; Mardinoglu, Adil
Background and Objectives: Favipiravir (FPV) is an antiviral medication and has an inhibitory effect on Cytochrome P450 (CYP2C8) protein, which is mainly involved in drug metabolism in the liver, and the expression of this gene is known to be enhanced in neuronal cells. The metabolization of Paclitaxel (PTX), a chemotherapeutic drug used in cancer patients, was analyzed for the first time in the human SH-SY5Y neuroblastoma cell line for monitoring possible synergistic effects when administered with FPV. Materials and Methods: Further, in vitro cytotoxic and genotoxic evaluations of FPV and PTX were also performed using wide concentration ranges in a human fibroblast cell culture (HDFa). Nuclear abnormalities were examined under a fluorescent microscope using the Hoechst 33258 fluorescent staining technique. In addition, the synergistic effects of these two drugs on cultured SH-SY5Y cells were determined by MTT cell viability assay. In addition, the death mechanisms that can occur in SHSY-5Y were revealed by using the flow cytometry technique. Results: Cell viability analyses on the HDFa healthy cell culture showed that both FPV and PTX have inhibitory effects at higher concentrations. On the other hand, there were no significant differences in nuclear abnormality numbers when both of the compounds were applied together. Cell viability analyses showed that FPV and PTX applications have higher cytotoxicity, which indicated synergistic toxicity against the SHSY-5Y cell line. Also, PTX exhibited higher anticancer properties against the neuroblastoma cell line when applied with FPV, as shown in both cytotoxicity and flow cytometry analyses. Conclusions: In light of our findings, the anticancer properties of PTX can be enhanced when the drug application is coupled with FPV exposure. Moreover, these results put forth that the anticancer drug dosage should be evaluated carefully in cancer patients who take COVID-19 treatment with FPV.
An Early and Accurate Diagnosis and Detection of the Coronary Heart Disease Using Deep Learning and Machine Learning Algorithms
(Springer Nature, 2025) Demir, Seda; Selvitopi, Harun; Selvitopi, Zulkuf
This study provides an extensive analysis of the role of Machine Learning (ML) and Deep Learning (DL) techniques in the early diagnosis of Coronary Heart Disease (CHD), one of the primary causes of cardiovascular morbidity and mortality worldwide. Early diagnosis is crucial to slow disease progression, prevent severe complications such as heart attacks, and enable timely interventions. We examine the impact of dataset variability on model performance by applying various ML and DL algorithms, including Multilayer Perceptron (MLP), Artificial Neural Networks (ANN), Convolutional Neural Network (CNN), Long Short-Term Memory (LSTM), Support Machine Vector (SVM), Logistic Regression (LR), Decision Tree (DT), k-Nearest Neighbor (kNN), Categorical Naive Bayes (CategoricalNB), and Extreme Gradient Boosting (XGBclassifier) to two distinct datasets: the comprehensive Framingham dataset and the UCI Heart Disease dataset. Before model training, data preprocessing techniques such as Hotdecking, Synthetic Minority Oversampling Technique (SMOTE), and normalization were implemented to enhance data quality. Model performance was evaluated using a range of metrics, including accuracy, precision, recall, F1-score, and area under the curve (AUC). The results reveal that the SVM model achieved the highest accuracy of 92.42%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$92.42\%$$\end{document} on the UCI dataset, while XGBclassifier attained the highest accuracy of 90.97%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$90.97\%$$\end{document} on the Framingham dataset, surpassing the performance reported in existing literature. These findings emphasize the potential of ML and DL methods for the early diagnosis of CHD and demonstrate the significance of dataset selection on model performance. This study offers valuable insights into the effectiveness of ML and DL approaches, underscoring the importance of data-driven strategies in advancing healthcare for the early detection and management of CHD and similar cardiovascular diseases.
Enhanced Photodiode Performance: Au/Boron-dipyrromethene Structure Unveiling High Photosensitivity and Efficiency
(Elsevier Sci Ltd, 2025) Duman, Songul; Gul, Elif Yildiz; Aydemir, Murat; Selvitopi, Harun; Ozer, Tuba Oznuluer; Ecik, Esra Tanriverdi
In this study, the classical boron-dipyrromethene (4,4- difuoro-4-borata-3a-azonia-4a-aza-s-indacene, BODIPY) unit (C1), its halogenated derivative (C2), and disytryl derivative (C3), which are known for their absorption properties in the near-infrared (NIR) region, were synthesized. Chemical and photophysical investigations were carried out to reveal the complex features of the molecules using a range of spectroscopic techniques and their energies were calculated by Density Functional Theory (DFT) with the same basis set. The BODIPY thin films were coated on n-Si substrates using a spin coater to examine the effect of interlayer on photodiode/photovoltaic properties and Au/BODIPY (C1, C2, C3)/n-Si/Ag photodiodes having rectifying behavior were fabricated. The thermionic emission theory (TET) and modified Norde's functions were utilized to determine the rectification ratio, ideality factor, series resistance, and barrier height values of diodes. The values of barrier height, ideality factor, and rectification ratio at +/- 2 V were calculated as 0.81 eV, 1.83 and 2.47x103 for the Au/C1/n-Si/Ag diode, 0.83 eV, 1.63, and 7.06 x103 for the Au/C2/n-Si/Ag diode and 0.84 eV, 1.51 and 2.14x104 for the Au/C3/ n-Si/Ag diode in the dark from the TET, respectively. The values of FF (%) and efficiency were calculated as 38 and 1.51 for the Au/C1/n-Si/Ag diode, 47 and 2.03 for the Au/C2/n-Si/Ag diode, and 56 and 3.68 for the Au/ C3/n-Si/Ag diode under 100 mW/cm2 illumination, respectively. With remarkable electrical properties, the Au/ C3/n-Si/Ag photodiode stood out among the others, performing better in both dark and light conditions. Therefore, the produced diodes based on the BODIPY thin film can be employed as photodiodes/photovoltaics in optoelectronic applications.
Exploring the Potential of Anthracene Derivatives as Fluorescence Emitters for Biomedical Applications
(Royal Society of Chemistry, 2023) Aydemir, Murat; Haykir, Gulcin; Selvitopi, Harun; Yildirim, Ozge Caglar; Arslan, Mehmet Enes; Abay, Bahattin; Turksoy, Figen
Two novel anthracene derivatives were synthesized, and detailed photo-physical and biological investigations were carried out using a variety of spectroscopy techniques. The effect of cyano (-CN) substitution was found to be effective to alter the charge population and frontier orbital energy levels via Density Functional Theory (DFT) calculations. Particularly, the introduction of styryl and triphenylamine groups attached to the anthracene core helped to increase the conjugation relative to the anthracene moiety. The results revealed that the molecules have intramolecular charge transfer (ICT) properties, occurring from the electron donating triphenylamine to the electron accepting anthracene moiety in solutions. In addition, the photo-physical properties are strongly cyano-dependent, where the cyano-substituted (E/Z)-(2-anthracen-9-yl)-3-(4 '-(diphenylamino)biphenyl-4yl)acrylonitrile molecule showed stronger electron affinity due to the enhanced internal steric hindrance compared to the (E)-4 '-(2-(anthracen-9-yl)vinyl)-N,N-diphenylbiphenyl-4-amine molecule, which resulted in a lower photoluminescence quantum yield (PLQY) value and a shorter lifetime in the molecule. Besides, the Molecular Docking approach was used to investigate possible cellular staining targets to confirm cellular imaging potential of the compounds. Moreover, cell viability analyses put forth that synthesized molecules do not exhibit significant cytotoxicity under 125 mu g mL(-1) concentration on the human dermal fibroblast cell line (HDFa). Moreover, both of the compounds showed great potential in cellular imaging of HDFa cells. Compared to Hoechst 33258, a common fluorescent dye used for nuclear staining, the compounds showed higher magnification of cellular structure imaging capacity by staining the whole cellular compartment. On the other hand, bacterial staining showed that ethidium bromide has higher resolution in monitoring Staphylococcus aureus (S. aureus) cell culture.
Finite Difference/Finite Element Simulation of the Two-Dimensional Linear and Nonlinear Higgs Boson Equation in the De Sitter Space-Time
(Springer, 2022) Selvitopi, Harun
In this work, finite element simulation of the two-dimensional linear and nonlinear form of the Higgs boson equation in de Sitter space-time is presented. The mathematical model of the problem is linear and power type nonlinear Klein-Gordon-like partial differential equations. Therefore, we discretize the temporal variable using the finite difference method and we also discretize the spatial variable using the finite element method. We use the Newton linearization technique which is one of the most useful linearization techniques for the linearization of the nonlinear partial differential equations. In the Newton method, we consider the Jacobian matrix numerically. Applying the considered numerical scheme we obtain the Bubble-like solutions in good agreement with the numerical results and theory available in the literature.
Glioma Beyin Tümörü Modelinin Fizik Bilgili Sinir Ağı ile Çözümü
(2025) Şenlik, Berrin; Selvitopi, Harun
Yapılan çalışmaları incelediğimizde görüyoruz ki, tıp ve biyoloji alanındaki problemleri Fizik Bilgili Sinir Ağı Yöntemi (FBSA) ile matematiksel olarak modellenmektedir. İnsanların hayatını tehdit eden beyin tümörlerinin erken tanı ve teşhisi için FBSA kullanılmıştır. Bu çalışmada beyin tümörü hücrelerini zaman ve konum olarak yayılımını ifade eden doğrusal olmayan kısmi diferansiyel denklemleri temel alarak problemleri bir ve iki boyutta FBSA çözümleri ele alınmıştır.
Novel Styryl-Thiazole Hybrids as Potential Anti-Alzheimer's Agents
(Royal Society of Chemistry, 2023) Gouleni, Niki; Di Rienzo, Annalisa; Yilmaz, Ahmet; Selvitopi, Harun; Arslan, Mehmet Enes; Mardinoglu, Adil; Cacciatore, Ivana
In this study, combining the thiazole and cinnamoyl groups into the styryl-thiazole scaffold, a series of novel styryl-thiazole hybrids (6a-p) was rationally designed, synthesized, and evaluated by the multi-target-directed ligands strategy as potential candidates for the treatment of Alzheimer's disease (AD). Hybrids 6e and 6i are the most promising among the synthesized hybrids since they are able to significantly increase cell viabilities in A beta 1-42-exposed-human neuroblastoma cell line (6i at the concentration of 50 mu g mL-1 and 6e at the concentration of 25 mu g mL-1 resulted in similar to 34% and similar to 30% increase in cell viabilities, respectively). Compounds 6e and 6i exhibit highly AChE inhibitory properties in the experimental AD model at 375.6 +/- 18.425 mU mL-1 and 397.6 +/- 32.152 mU mL-1, respectively. Moreover, these data were also confirmed by docking studies and in vitro enzyme inhibition assays. Compared to hybrid 6e and according to the results, 6i also has the highest potential against A beta 1-42 aggregation with over 80% preventive activity. The in silico prediction of the physicochemical properties confirms that 6i possesses a better profile compared to 6e. Therefore, compound 6i presents a promising multi-targeted active molecular profile for treating AD considering the multifactorial nature of AD, and it is reasonable to deepen its mechanisms of action in an in vivo experimental model of AD. Novel styryl-thiazole hybrids (6a-p) was rationally designed, synthesized, and evaluated by the multi-target-directed ligands strategy as potential candidates for the treatment of Alzheimer's disease.
Numerical Investigation of Damped Wave Type MHD Flow with Time-Varied External Magnetic Field
(Elsevier, 2022) Selvitopi, Harun
In recent years, the MHD flow problems has received a significant attention as the behavior of the flow changes when the Lorentz force (electromagnetic force) acts on the fluid. The aim of this study is to observe numerically the behavior of damped wave-type magnetohydrodynamic (MHD) flow in a sufficiently long rectangular channel having time-varied oblique magnetic field. The mathematical model of the considered problem is coupled convection-diffusion heat-type for the velocity of the fluid and convection-diffusion wave-like for the induced magnetic field. To obtain the numerical simulation, we have used the finite difference method for the discretization of the temporal variable and we also have considered the Galerkin finite element method for the discretization of the spatial variable. In the numerical investigation procedure, the time-varied oblique magnetic field has been considered several functions of the time as polynomial, exponential, and trigonometric e.t.c. We have also considered the several values of the Hartmann number and several directions of the induced magnetic field. In literature, there are many studies about numerical solution of steady and unsteady magnetohydrodynamic flow problems using finite element method, boundary element method and finite difference method e.t.c. For all that, it is the first time in the literature, the damped wave type MHD flow has been considered numerically in this study. The numerical results show Hartmann layer occurs for the high Hartmann numbers in accordance with the nature of the MHD flow. We have obtained numerical results in good agreement with the numerical results available in the literature. The acquired numerical results have been presented by the contour plots.
Numerical Investigation of the Fully Damped Wave-Type Magnetohydrodynamic Flow Problem
(MDPI, 2024) Demir, Seda; Selvitopi, Harun
Magnetohydrodynamic (MHD) flow plays a crucial role in various applications, ranging from nuclear fusion devices to MHD pumps. The mathematical modeling of such flows involves convection-diffusion-type equations, with fluid velocity governed by the Navier-Stokes equations and the magnetic field determined by Maxwell's equations through Ohm's law. Due to the complexity of these models, most studies on steady and unsteady MHD equations rely on numerical methods, as theoretical solutions are limited to specific cases. In this research, we propose a damped-wave-type mathematical model to describe fluid flow within a channel, taking into account both the velocity and magnetic field components. The model is solved numerically using the finite difference method for time discretization and the finite element method for spatial discretization. Numerical results are displayed graphically for different values of Hartmann numbers, and a detailed analysis and discussion of the solutions are provided.
Numerical Methods for the Multiplicative Partial Differential Equations
(de Gruyter Open Ltd, 2017) Yazici, Muhammet; Selvitopi, Harun
We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.
Numerical Results for the Klein-Gordon Equation in De Sitter Spacetime
(Wiley, 2019) Selvitopi, Harun; Yazici, Muhammet
We consider the initial value problem for the Klein-Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank-Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.
Stabilized FEM Solution of Magnetohydrodynamic Flow in Different Geometries
(2022) Selvitopi, Harun
In this study, the stable numerical solution of the magnetohydrodynamic (MHD) flow in different geometries is presented using the stabilized finite element method (FEM). Numerical solution of coupled convection-diffusion type MHD equations have been acquired for the different Hartmann numbers (M_i) and different angles of the MHD flows. The resultant matrix-vector system has been solved as a whole with the reciprocal MHD flow and boundary conditions. We have observed from the solution of reciprocal MHD flow when the Hartmann number increases the velocity and the induced magnetic field of the flows decrease. We have been acquired the stable numerical solution for the M_i=〖10〗^2 Hartmann number. The obtained stable numerical results are displayed by graphics.
Zamana Bağlı Konvektif Katsayılı Kararsız Konveksiyon-difüzyon-reaksiyon Denkleminin Sonlu Elemanlar Yöntemi ile Çözümü
(2022) Kuzulugil, Sümeyra Didem; Selvitopi, Harun
Bilindiği üzere, birçok mühendislik ve fizik problemleri konveksiyon-difüzyon-reaksiyon denklemi ile matematiksel olarak modellenmektedir. Literatürde, bahsi geçen problemle ilgili sayısal ve teorik farklı çalışmalar mevcuttur. Bu tez çalışmasında ise zamana bağlı konvektif katsayılı konveksiyon-difüzyon-reaksiyon denklemi Crank-Nicolson/sonlu elemanlar hibrit sayısal yöntemi ile çözülmüştür. Sayısal çözüm elde edilirken Fortran programlama dili kullanılmıştır. Sayısal sonuçlar farklı difüzyon ve zamana bağlı konvektif katsayıları için farklı zaman adımlarında elde edilmiş, incelenmiş ve yorumlanmıştır.