University of TabrizComputational Methods for Differential Equations2345-398213420251001Transformation of motion equations in the oil extraction process into Roesser-type equations and their solution using the Laplace transform108510942015210.22034/cmde.2025.62544.2766ENFikret A.AlievInstitute of Applied Mathematics, Baku State University, Baku, Azerbaijan.Reshad M.TagiyevAzerbaijan State Oil and Industry University, Baku, Azerbaijan.Orkhan Z.NamazovSumgayit State University, Sumgayit, Azerbaijan.Amankeldy K.TurarovEast Kazakhstan State Technical University, Ust-Kamenogorsk, Kazakhstan.Journal Article20240718In the article, the method of solving Roesser-type equations with the Laplace transformation for modeling and analysis of oil extraction processes is analyzed. Roesser-type equations are widely used to describe the dynamics of multidimensional systems, and their solution is important in improving the efficiency of oil extraction processes. First, the structure and properties of Roesser-type equations are presented. Later, the process of solving these equations is presented step by step by applying the Laplace transform method. By converting the special differential equations given by this method to simpler algebraic equations, both analytical and computer calculations can be significantly simplified. Based on the calculations and examples, it is shown that the proposed method provides high accuracy and efficiency. The research results enable the application of new approaches in the optimization and management of oil extraction processes.https://cmde.tabrizu.ac.ir/article_20152_1ccf498eba275f3a63c3a8405e924ae5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001ADI numerical method for modeling stock insurance based on spread options109511011875810.22034/cmde.2024.59333.2524ENReyhaneMohamadinegadDepartment of Mathematics, Faculty of Mathematics Science and Computer, Allameh Tabataba’i University, Tehran, Iran.AbdolsadehNeisyDepartment of Mathematics, Faculty of Mathematics Science and Computer, Allameh Tabataba’i University, Tehran, Iran.JafarBiazarDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.Journal Article20231124This paper introduces a spread option model based on two underlying assets, namely Bandar Abbas oil refining (Shebandar) and Tehran oil refining (Shatran) companies. Regarding the available data of the former, we propose the jump-diffusion model for its dynamics. After constructing our portfolio, we first consider a partial integro-differential equation (PIDE) for the spread option model. Then, by making some alterations to the literature of the problem and parameters of the model, it is demonstrated that the assumed option can be considered as insurance, hedging the stocks mentioned above. The PIDE is solved by the well-known ADI numerical method. Finally, we utilize real data extracted from the Tehran Stock Exchange, and a reliable result is obtained by using MATLAB software.https://cmde.tabrizu.ac.ir/article_18758_116794c2a5689e0fabe1bb23830eb7cc.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Spectral collocation algorithm for the fractional Bratu equation via Hexic shifted Chebyshev polynomials110211161854810.22034/cmde.2024.61045.2621ENAhmed GamalAttaDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt.Jomana FaragSolimanDepartment of Mathematics, Faculty of Science, Galala University, Egypt.Elaf WaelElsaeedDepartment of Mathematics, Faculty of Science, Galala University, Egypt.Mostafa WaelElsaeedDepartment of Mathematics, Faculty of Science, Galala University, Egypt.Youssri HassanYoussriDepartment of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.Journal Article20240330This paper offers a numerical collocation scheme for solving the fractional nonlinear Bratu differential equation. We obtain a system of nonlinear equations using our spectral collocation method, which we then solve iteratively using Newton’s method to obtain an approximate solution. Additionally, numerical comparisons are made between the proposed strategy and several numerical strategies documented in various literature. The numerical findings verify the accuracy, computational efficiency, and ease of use of the recommended approachhttps://cmde.tabrizu.ac.ir/article_18548_da542ee8bccee2b50a73b527aa8bc807.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Symmetries and conservation laws of the Berger metric on a squashed three-sphere111711241893110.22034/cmde.2024.56893.2381ENYadollahAryaNejadDepartment of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.MasomehPadiz FoumaniDepartment of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.Journal Article20230529 In this work, we obtain Noether, Lie, and Killing symmetries of the Lagrangian of the Berger metric on a squashed three-sphere. With the help of the result of Noether’s theorem, we have presented the expressions for conservation laws corresponding to all Noether symmetries.https://cmde.tabrizu.ac.ir/article_18931_5583389e4ffba2ef8107c5d24f4d7a93.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Hyers-Ulam and exponential stabilities of autonomous and non-autonomous difference equations112511341837510.22034/cmde.2024.59702.2544ENGulRahmatDepartment of Mathematics, Islamia College Peshawar, Pakistan.AfaqAhmadDepartment of Mathematics, Islamia College Peshawar, Pakistan.MuhammadSarwar1. Department of Mathematics, University of Malakand, Dir Lower, Pakistan.\\
2. Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.KamaleldinAbodayehDepartment of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.CemilTunçSchool of Engineering and Natural Sciences, Istanbul Medipol University, 34810, Istanbul, Turkey.0000-0003-2909-8753Journal Article20231223 In this manuscript, we studied the Hyers-Ulam and exponential stabilities of autonomous and non-autonomous difference equations of first and second order. Ultimately, we provide some examples to support our results.https://cmde.tabrizu.ac.ir/article_18375_2e80d047007ef9f3e23b57d339e58eeb.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A Chebyshev wavelet approach to the generalized time-fractional Burgers-Fisher equation113511471865510.22034/cmde.2024.61020.2617ENNasserAghazadeh1. Department of Mathematics, Izmir Institute of Technology, Izmir, Türkiye.
2. Center for Theoretical Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan.0000-0003-2705-8942Journal Article20240327This work proposes a new method for obtaining the approximate solution of the time-fractional generalized Burgers Fisher equation. The method’s main idea is based on converting the nonlinear partial differential equation to a linear partial differential equation using the Picard iteration method. Then, the second kind Chebyshev wavelet collocation method is used to solve the linear equation obtained in the previous step. The technique is called the Chebyshev Wavelet Picard Method (CWPM). The proposed method successfully solves the time fractional generalized Burgers-Fisher equation. The obtained numerical results are compared with the exact solutions and with the solutions obtained using the Haar wavelet Picard method and the homotopy perturbation method.https://cmde.tabrizu.ac.ir/article_18655_f72a51a84c03615e66a3705fa103aa4f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Zero-Hopf bifurcation in a four-dimensional quartic polynomial differential system via the averaging theory of the third order114811611861010.22034/cmde.2024.61831.2690ENChamseddineBouazizDepartment of Mathematics, University of Annaba, Laboratory LMA P.O. Box.12;23 Annaba, Algeria.AmarMakhloufDepartment of Mathematics, University of Annaba, Laboratory LMA P.O. Box.12;23 Annaba, Algeria.Achref EddineTabetDepartment of Mathematics, University of Annaba, Laboratory LMA P.O. Box.12;23 Annaba, Algeria.Journal Article20240527 The averaging theory of third order shows that for a 4-dimensional Quartic Polynomial Differential System, at most 36 limit cycles can bifurcate from one singularity with eigenvalues of the form ±ωi, 0, and 0.https://cmde.tabrizu.ac.ir/article_18610_e0be24d179eb4dc75070af6670d78f98.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Numerical multiscale methods to determine the coefficient in diffusion problems116211761902710.22034/cmde.2024.59745.2547ENMarziehTavakolianDepartment of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.AliHatamDepartment of Applied Mathematics, Amirkabir University of Technology, No. 350, Hafez Ave, Valiasr Square, Tehran, Iran 1591634311.0000-0002-3743-8589MortezaFotouhiDepartment of Mathematical Sciences, Sharif University of Technology, Tehran 11365-9415, Iran.EdmundChadwickSchool of Science, Engineering & Environment, University of Salford, Salford, M5 4WT, UK.0000-0002-3743-8589Journal Article20231224Here we study the inverse problem of determining the highly oscillatory coefficient $a^\varepsilon$ in some PDEs of the form $ u^\varepsilon_t - \nabla. (a^\varepsilon(x) \nabla u^\varepsilon)=0$, in a bounded domain $\Omega \subset\mathbb{R}^d $; $\varepsilon$ indicates the smallest characteristic wavelength in the problem ($0 < \varepsilon \ll 1$).<br />Assume that $g(t, x)$ is given input data for $(t, x) \in (0,T) \times\partial \Omega$ and the associated output is the thermal flux $a^\varepsilon(x)\nabla u(T_0,x)\cdot n(x)$ measured on the boundary at a given time $T_0$. Due to the ill-posedness of the inverse problem, we reduce the dimension by seeking effective parameters. For the forward solver, we apply either analytic homogenization or some numerical multiscale methods such as the FE-HMM and LOD method.https://cmde.tabrizu.ac.ir/article_19027_56194d4c4aca68283a77140922eaf2fa.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A novel framework for Breast cancer scoring based on machine learning techniques using Immunohistochemistry images117711881859410.22034/cmde.2024.63113.2810ENHasanain HayderRazzaq1. Faculty of Computer Science and Information Technology, University Tun Hussein Onn Malaysia, Batu Pahat, 86400, Johor, Malaysia.
2. College of Medicine, Jabir ibn Hayyan Medical University, Najaf, Iraq.RozaidaGhazaliFaculty of Computer Science and Information Technology, University Tun Hussein Onn Malaysia, Batu Pahat, 86400, Johor, Malaysia.Loay E.GeorgeUniversity of Information Technology and Communication (UoITC), 10001, Baghdad, Iraq.MuhammadZulqarnainDepartment of Computer Science & IT, Cholistan University of Veterinary and Animal Sciences, Bahawalpur, Punjab, Pakistan.Journal Article20240824Breast cancer is classified as a serious disease in the medical field, and there is no doubt that breast cancer detection requires effective and accurate techniques. Integrating deep learning (DL) and machine learning (ML) methods has shown promising results in this area. In this research, we introduce a hybrid approach for breast cancer diagnosis, which is centered on the analysis of immunohistochemical images. The proposed method encompasses algorithms for image pre-processing, segmentation, extracting informative indicators (such as relative cell area and intensity), and an algorithm for categorizing the molecular harmonic subtype of breast cancer. The number of the sample was 598, divided into training 70% and testing 30%. The 5-fold cross-validation was used to assess the proposed approach. Experimental results showcased the effectiveness of the proposed hybrid method in achieving superior performance in the detection of breast cancer, especially within breast cancer scoring systems. The accuracy of our proposed approach, which involved combining HSV integration with adaptive high boost filtering, reaches a peak at 96.5% when using SVC (linear kernel). Moreover, the precision, recall, F1-score, and specificity metrics are recorded at 95.29%, 99.99%, 95.59%, and 99.28%, respectively. Additionally, this study evaluated the efficacy of the proposed model in comparison to various other traditional breast cancer detection approaches.https://cmde.tabrizu.ac.ir/article_18594_561ddc215b5ba03d8fdfe86024494571.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Numerical solutions for a branch crack in a half-plane118912001899710.22034/cmde.2024.58971.2499ENRaziyehGhorbanpoorEtrat School, Department of Education of South Khorasan, Namjo Street, Qaenat, 100190 South Khorasan, Iran.Nik Mohd AsriNik LongDepartment of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia.Mohammad HasanAbdiDepartment of Physics, Qaenat Branch, Islamic Azad University, Qaenat, Iran.Journal Article20231024The branch crack subjected to a remote stress in a half-plane of elasticity is modeled using singular integral equations (SIE) based on the distributed dislocation and complex potential method. Numerical solution to the obtained SIE is discovered using the appropriate quadrature formulas. Numerical works exhibit the nature of stress intensity factors (SIF) for each branch.https://cmde.tabrizu.ac.ir/article_18997_bcafae60ce83b068cda13a242698264e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Semi-analytical solutions for time-fractional Cauchy reaction-diffusion equations via the new Elazki transform iterative method120112151854410.22034/cmde.2024.53470.2253ENShivaji AshokTarateDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.Ashok PBhadaneDepartment of Mathematics, Loknete Vyankatrao Hiray Arts, Science and Commerce College, Nashik, Maharashtra, India.Shrikisan BGaikwadDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.Kishor AshokKshirsagarDepartment of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.Journal Article20220914In this article, the estimated analytic solutions for time-fractional Cauchy Reaction-Diffusion Equations (CRDE) are obtained using a New Elzaki Transform Iterative Method (NETIM). This method is the fusion of the Elzaki transform and the Iterative approach. The proposed technique is elegant and easy to adopt and comprehend. The semi-analytical results demonstrate, as this paper shows, a graphical interpretation of the solution using the mathematical software “Mathematica Wolform” and considering Caputo’s sense derivatives to analytical results, the suggested strategy is efficient and straightforwardhttps://cmde.tabrizu.ac.ir/article_18544_b9a08c4ca73a8849791515248f8ce757.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A novel Bernstein operational matrix: applications for conformable fractional calculus121612321907710.22034/cmde.2024.61944.2700ENMohsenAlipourDepartment of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, Iran.Journal Article20240603In this work, we focus on the conformable fractional integral and derivative. We approximate the one and two-variable functions by the Bernstein basis and its dual basis while studying convergence. Then, we get the new operational matrix for conformable fractional integral based on the Bernstein basis. To show the effectiveness of these approximations and conformable integral operational matrix, we apply them for solving the nonlinear system of differential equations, the optimal control problem in the conformable fractional sense, and the space conformable fractional telegraph equation.https://cmde.tabrizu.ac.ir/article_19077_97e37ef0f2ced9d378c70f945aa2b594.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Application of the analytical method for solving the chemical kinetics system123312491939510.22034/cmde.2024.64302.2910ENInaam RikanHassanUniversity of Information Technology and Communications, (UoITC), Baghdad, Iraq.Journal Article20241101This work introduces an enhanced $tan(\chi/2)$-expansion method to obtain exact solutions for<br />chemical kinetics systems. This technique works directly with the governing equations of chemical kinetics systems. Our work yields many new fundamental traveling wave solutions that combine periodic functions with soliton-like and other trigonometric shapes. To better illustrate our solutions, we show visual representations by assigning specific values to the arbitrary constants. The improved expansion method successfully obtains kink, singular kink, and multiple soliton solutions. The results demonstrate that the method is effective for real-world applications and physics equations. Graphical visualizations support our findings to show the method's accuracy and reliability. Our suggested method effectively solves nonlinear equations and provides useful results for studying complicated wave behavior across multiple scientific disciplines.https://cmde.tabrizu.ac.ir/article_19395_6ff5fa019484c5dca60354feee0b8059.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001The $(4\alpha-\rho)$ order Sturm-Liouville problem with generalized fractional derivative125012591921110.22034/cmde.2024.60925.2604ENMohammadJafariDepartment of Science, Payame Noor University, Tehran, Iran.FarhadDastmalchi SaeiDepartment of Mathematics Faculty of Science Tabriz Branch, Islamic Azad University, Tabriz, Iran.Journal Article20240315In this paper, we analyze the generalized fractional derivative with two parameters for fourth-order Sturm-Liouville problems. These parameters are $\alpha$({\em the fractional order}) and $\rho$ ({\em a real number}). In the following, we discuss five different forms of Sturm--Liouville problems, which are solved using the $\rho-$Laplace transform.https://cmde.tabrizu.ac.ir/article_19211_7235e1e82b6c4b9c81c017aef21cbd3a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Dynamics and bifurcation control of a fractional-order delayed predator-prey model with an omnivore126012781861110.22034/cmde.2024.61871.2696ENAbdul HussainSuroshDepartment of Mathematics, Baghlan University, Pol-e-Khomri, Baghlan, Afghanistan.RezaKhoshsiar GhazianiDepartment of Applied Mathematics, Shahrekord University, Shahrekord, P.O. Box 115, Iran.JavadAlidoostiDepartment of Applied Mathematics, Shahrekord University, Shahrekord, P.O. Box 115, Iran.Journal Article20240529In this study, we propose a novel fractional delayed predator-prey model that includes an omnivorous species and explore bifurcation control through a state feedback control strategy. We begin by deriving the characteristic polynomial using the Laplace transform and establish new sufficient conditions for stability analysis and Hopf bifurcation, treating the time delay $ \tau$ as a bifurcation parameter. To address the Hopf bifurcation in the uncontrolled system, we design a state feedback controller with a time delay. Our results indicate that the time delay $ \tau$ significantly affects the onset of the Hopf bifurcation. Additionally, the inclusion of a fractional order $ 0<\alpha \leq 1 $ enhances solution stability while adding complexity to the dynamics of the model. We find that judicious selection of the feedback gain can delay bifurcation, highlighting the critical role of control effort. To validate our theoretical findings, we present numerical simulations conducted using a modified Adams-Bashforth-Moulton predictor-corrector method. These simulations support our theoretical results and demonstrate the efficacy of our proposed control strategy in managing the dynamical behaviors of the model.https://cmde.tabrizu.ac.ir/article_18611_ee996151774c5934d92fc106a7775699.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A new algorithm for indoor robot localization using the Turning function127912881891010.22034/cmde.2024.64173.2895ENPardisSadatian MoghaddamDepartment of Computer Science, Georgia State University, Atlanta, Georgia.AliVaziriDepartment of Systems and Enterprises, Stevens Institute of Technology, Hoboken, NJ, USA.AnitaErshadi OskoueiDepartment of Systems and Enterprises, Stevens Institute of Technology, Hoboken, NJ, USA.Journal Article20241027One of the complex challenges in the field of artificial intelligence and robotics is enabling robots to accurately determine their position. While this task is simpler in open spaces using antennas, satellites, and other tools, it becomes much more challenging in enclosed environments. Various methods are employed for indoor positioning, one of which involves low-cost rangefinders and polygon mapping around the robot.<br />This paper presents a new algorithm entitled RLuTF using turning functions and their geometric properties, allowing the robot to determine its position at any given moment.https://cmde.tabrizu.ac.ir/article_18910_c94dfb2479552a14e0bbba6a596b9f71.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A new three-step optimal without memory iterative scheme for solving non-linear equations with basins of attraction128913051848910.22034/cmde.2024.59161.2514ENShahidAdullahSchool of Advanced Sciences and Languages, VIT Bhopal University, Kothri-Kalan, Sehore, 466114, MP, India.NehaChoubeySchool of Advanced Sciences and Languages, VIT Bhopal University, Kothri-Kalan, Sehore, 466114, MP, India.SureshDaraSchool of Advanced Sciences and Languages, VIT Bhopal University, Kothri-Kalan, Sehore, 466114, MP, India.Journal Article20231110The primary focus of this study is to introduce a new three-step iterative method without memory for root finding by merging two different existing techniques. Based on the computational cost, the proposed method acquires optimal eight-order convergence with four functional evaluations (three evaluations for the function and one computation of the first derivative). Furthermore, the suggested scheme supports Kung-Traub’s Conjecture with an efficiency index of $8^\frac{1}{4}=1.682$. We also established the convergence criteria developed for the root-finding technique and demonstrated the fact that the suggested approach is eighth-order convergent. In order to demonstrate the efficacy as well as application of the constructed root-finding technique, we addressed a few practical engineering models and some non-linear functions. In contrast to several existing approaches, this particular method converges more quickly. Finally, several forms of complex functions are taken into consideration under basins of attraction in order to observe the overall fractal behavior of the proposed technique.https://cmde.tabrizu.ac.ir/article_18489_3f20f621ef734b40426be4c60048dfb9.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A regularization technique to overcome the Ill-posedness arising in specific engineering models: formulation, implementation, error analysis, and some engineering applications130613352008010.22034/cmde.2025.67540.3223ENSaeedHatamzadehDepartment of Electrical and Electronics Enginnering, Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey.TofighAllahviranlooResearch Center of Performance and Productivity Analysis, Istinye University, Istanbul, Turkey.ZahraMasouriDepartment of Mathematics, Isl.C., Islamic Azad University, Islamshahr, Iran.Journal Article20250527This article presents a regularization technique for the stable numerical solution of first-kind Fredholm integral equations, which frequently arise in the mathematical modeling of engineering and physical science problems. The proposed technique combines an approximation framework based on a special representation of the triangular function vector forms and their properties with a stabilization strategy to convert the original ill-posed problem into a well-posed algebraic system. Another notable advantage is the reduced computational cost of the proposed technique, as it eliminates the need for performing any integrations during the setup of the algebraic system. Detailed error analysis and convergence proofs are provided, offering rigorous theoretical guarantees for the method’s performance. Numerical experiments on test problems demonstrate the efficiency, stability, and high accuracy of the proposed technique, especially when compared with other regularization methods. Furthermore, the proposed technique is applied to analyze some engineering models, including electromagnetic scatterers and thin-wire antennas. In all cases, the results show excellent agreement with full-wave simulations performed using Altair FEKO software. These findings confirm the robustness, versatility, and computational effectiveness of the proposed regularization strategy for practical ill-posed problems.https://cmde.tabrizu.ac.ir/article_20080_048b039e2967e566d84555d86058a1a1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Computational applications by ITEM and the variational method for solving the Hamiltonian amplitude equation133613561851410.22034/cmde.2024.62097.2714ENElvirAkhmetshinDepartment of Economics and Management of Elabuga Institute, Kazan
Federal University, Kazan, Russia
Moscow Aviation Institute (National Research University), Moscow, Russia.IlyosAbdullayevDepartment of Management and Marketing, Urgench State
University, Urgench, Uzbekistan.NalbiyTuguzDepartment of Higher Mathematics, Kuban State Agrarian
University
named after I.T. Trubilin, Krasnodar, Russia.DmitryFugarovDepartment of Automation and Mathematical Modeling in the Oil
and Gas Industry, Don State Technical University, Rostov-on-Don,
Russia.DianaStepanovaHigher School of Finance, Plekhanov Russian University of Economics, Moscow, Russia.Journal Article20240613The paper presents a significant improvement to the implementation of the improved $\tan(\phi(\xi)/2)$-expansion method (ITEM) for solving the Hamiltonian amplitude equation (HAE). We seek to improve the exact solutions by applying the ITEM. Computed solutions are compared with previously published results obtained using the simplest equation method \cite{Eslami} and the $(G'/G,1/G)$-expansion method \cite{Demiray}. There is clear evidence that the new approach produces results that are as good as, if not better than, published results determined using the other methods. The main advantage of the method is that it offers further solutions. By using this method, exact solutions, including the hyperbolic function solution, traveling wave solution, soliton solution, rational function solution, and periodic wave solution of this equation, have been obtained. Moreover, variational principles for the HAE are formulated. The invariance identities of the HAE involving the Lagrangian $L$ and the generators of the infinitesimal Lie group of transformations have been utilized for writing down their first integrals via Noether's theorem, Logan. We demonstrate the simplest example of the application of this technique, taking the box-shaped initial pulse and an ansatz based on linear Jost functions. We consider a combination of two boxes of opposite signs, the total area of the initial pulse being thus zero. Therewith, we develop a variational approximation for finding the eigenvalues of this pulse, by a piece-wise linear ansatz and tanh functions of the piece-wise linear function. Moreover, by using MATLAB, some graphical simulations were done to see the behavior of these solutions.https://cmde.tabrizu.ac.ir/article_18514_412a1dc0b18278ef4e76e9573c9af50c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Investigation of highly dispersive solitons for the concatenation model with power law nonlinearity using the improved modified extended tanh-function method135713661885310.22034/cmde.2024.61272.2632ENEmanSalahDepartment of Physics and Engineering Mathematics, Higher Institute of Engineering,El-Shorouk Academy, El-Shorouk, Cairo, Egypt.IslamSamirDepartment of Mathematics and Engineering Physics, Faculty of Engineering, Aim Shams University, Cairo, Egypt.Emad M.Abo El-DahabDepartment of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt.Hamdy M.AhmedDepartment of Physics and Engineering Mathematics, Higher Institute of Engineering,El-Shorouk Academy, El-Shorouk, Cairo, Egypt.MedhatAmmarDepartment of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt.MostafaEslamiDepartment of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.Journal Article20240418This research examines the phenomenon of optical solitons in the framework of the dispersive concatenation model, which incorporates three established models: the Lakshmanan-Porsezian-Daniel equation (LPDE), the Hirota equation (HE), and the nonlinear Schr¨odinger equation (NLSE). This model describes the soliton transmission dynamics across transcontinental and transoceanic dynamics. The model provided is situated within the context of nonlinear optics, a branch of optics that deals with optical phenomena in materials where the response of the medium to light is nonlinear. The equation appears to be a generalized model that combines several well-known equations from nonlinear optics. These equations often emerge as simplified descriptions of specific nonlinear effects in various optical systems. They capture phenomena like self-focusing, self-phase modulation, and soliton propagation, among others. The improved modified extended tanh scheme (IMETS) is utilized to derive solitons and other solutions for the investigated model. Many types of solutions are extracted with the help of the IMETS. These solutions include dark, bright, and singular solitons, as well as Weierstrass elliptic and singular periodic solutions. The nature of the extracted solutions is illustrated by introducing both 2D and 3D graphical representations and setting the parameters with appropriate values.https://cmde.tabrizu.ac.ir/article_18853_e8eeb95dff3412ad42b1b3ab4ba1530e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Study of p-Laplacian hybrid fractional differential equations involving the generalized Caputo proportional fractional derivative136713751855610.22034/cmde.2024.61552.2665ENSamiraZerbibLMACS Laboratory, Sultan Moulay Slimane University, Beni Mellal, Morocco.NajatChefnajLMACS Laboratory, Sultan Moulay Slimane University, Beni Mellal, Morocco.KhalidHilalLMACS Laboratory, Sultan Moulay Slimane University, Beni Mellal, Morocco.AhmedKajouniLMACS Laboratory, Sultan Moulay Slimane University, Beni Mellal, Morocco.Journal Article20240506In this paper, we investigate the existence of solutions for hybrid p-Laplacian differential equations involving the generalized fractional proportional Caputo derivative of order $1<\vartheta<2$, by employing Schauder’s fixed point theorem. To illustrate the practical application of our findings, we provide a concrete examplehttps://cmde.tabrizu.ac.ir/article_18556_018867e3cab684f9f2e9738656c19c79.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Computing high-index eigenvalues for the Sturm-Liouville equation with Robin boundary conditions137613841885010.22034/cmde.2024.60228.2566ENYasserKhaliliDepartment of Basic Sciences, Sari Agricultural Sciences and Natural Resources University, 578 Sari, Iran.MohammadDehghanDepartment of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.Journal Article20240123The calculation of the high-indexed eigenvalues of Sturm-Liouville problems tends to be a complex job. The larger the eigenvalues are estimated, the greater the scaled errors we gain. In this paper, we study the Sturm-Liouville problems subject to Rubin boundary conditions in which the high-indexed eigenvalues are computed by means of an efficient method. In contrast to previous methods, the estimated errors of further eigenvalues are less than the primary ones. A good illustration of the accuracy of our method can be delineated by some numerical examples.https://cmde.tabrizu.ac.ir/article_18850_edc94e5be345836f06eafea2a72f768c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001A higher-order orthogonal collocation technique for discontinuous two-dimensional problems with Neumann boundary conditions138513991851710.22034/cmde.2024.60344.2577ENSantosh KumarBhalDepartment of Mathematics, School of Advanced Sciences and Languages, Vellore Institute of Technology, Bhopal, India.Ashish KumarNandiDepartment of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai,
Tamil Nadu-600127, India.AbedallahRababahDepartment of Mathematical Sciences, United Arab Emirates University, United Arab Emirates.Journal Article20240130In this paper, the orthogonal spline collocation method (OSCM) is employed to address the solution of the Helmholtz equation in two-dimensional problems. It is characterized by discontinuous coefficients with certain wave numbers. The solution is approximated by employing distinct basis functions, namely, monomial along the x-direction and Hermite along the y-direction. Additionally, to solve the two-dimensional problems efficiently in the sense of computational cost with fewer operation counts, the matrix decomposition algorithm (MDA) is used to convert them into a set of one-dimensional problems. As a consequence, the resulting reduced matrix becomes non-singular in discrete cases. To assess the performance of the proposed numerical scheme, a grid refinement analysis is conducted to incorporate various wave coefficients of the Helmholtz equation. The illustrations and examples demonstrate a higher order of convergence compared to existing methods.https://cmde.tabrizu.ac.ir/article_18517_71ae40e11c30d19db7302764d1307b81.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001The study of maximal surfaces by Lie symmetry140014071885210.22034/cmde.2024.62234.2729ENAkramMohammadpouriFaculty of Mathematics, Statistics and Computer Sciences, University of Tabriz, Tabriz, Iran.SedighehHasannejadDepartment of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.AliHaji BadaliDepartment of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.Journal Article20240623Maximal surfaces, a fascinating class of surfaces in differential geometry, are identified by having a mean curvature equal to zero. This distinctive feature gives rise to a nonlinear second-order partial differential equation. In this current article, we delve into the symmetries that underlie the maximal surface equation. Next, we identify a one-dimensional optimal system of subalgebras that span these symmetries. It provides a powerful tool to analyze and manipulate the equation, making it easier to study. Finally, since we aim not only to explore the underlying symmetries of the maximal surface equation, we demonstrate how these symmetries can be harnessed to uncover and classify a wide range of maximal surfaces by using reduction methods.https://cmde.tabrizu.ac.ir/article_18852_d0b73cf79def7dbc53b007607703f582.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398213420251001Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity140814241843110.22034/cmde.2024.61349.2636ENDmitryFugarovDon State Technical University, Rostov-on-Don, Russia.AlexeyDengaevGubkin Russian State University of Oil and Gas, Moscow, Russia.IlyaDrozdovGubkin Russian State University of Oil and Gas, Moscow, Russia.VladimirShishulinGubkin Russian State University of Oil and Gas, Moscow, Russia.AnastasiyaOstrovskayaKuban State University, Krasnodar, Russia.Journal Article20240424In this paper, the improved $\tan\left(\Phi(\xi)/2\right)$-expansion method (ITEM) is proposed to obtain the fractional Biswas-Milovic equation. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions compared with other methods, such as [2]. Recently, this method has been developed for searching exact travelling wave solutions of nonlinear partial differential equations. These solutions might play an important role in nonlinear optics and physics. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving problems in nonlinear optics.https://cmde.tabrizu.ac.ir/article_18431_42400af11068e6e88ce8f2129fe40d50.pdf