University of TabrizComputational Methods for Differential Equations2345-398214220260401A mathematical and computational study of global stability in partially-ionized rotating plasma5025181958910.22034/cmde.2025.58316.2465ENVishalChandelDepartment of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, 177005, Himachal Pradesh, India.Sunil-Department of Mathematics & Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, 177005, Himachal Pradesh, India.PoonamSharmaDepartment of Mathematics, Govt. College Jawalaji, Jawalamukhi, Kangra, 176031, Himachal Pradesh, India.Journal Article20230906A mathematical and computational study of the impact of rotation on the thermal convection of partially-ionized plasma has been explored using both linear and nonlinear analyses. The method of normal mode analysis has been used to study the linear analysis whereas, for nonlinear analysis, we have used the generalized energy method. For numerical analysis, we have employed the Galerkin method. It has been found that the Rayleigh number for nonlinear analysis is the same as station ary convection. Hence, we concluded that there is no sub-critical region and the system is globally stable. The effect of collision plays an important role in the energy decay analysis. It has also been observed that for stationary convection, the collisional frequency has no impact on stability, whereas rotation stabilizes the system. The effect of various parameters has also been discussed for oscillatory convection. The stability characteristics for different bounding surfaces are examined. For low rotation rates, partially ionized plasma confined between rigid–rigid boundaries is the most stable configuration; however, at high rotation rates, the free–free bounding surfaces yield the greatest stability.https://cmde.tabrizu.ac.ir/article_19589_8af2ab5bc39e7052ff28497bcb79986a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Numerical approximation of coupled Schrödinger equations via NUAH B-spline DQM5195481940010.22034/cmde.2025.64656.2935ENMamtaKapoorMarwadi University Research Center, Department of Mathematics, Faculty of Engineering & Technology, Marwadi University, Rajkot, 360003, Gujarat, India.GeetaAroraDepartment of Mathematics, Lovely Professional University, Phagwara, Punjab, India-144411.VarunJoshiDepartment of Mathematics, Lovely Professional University, Phagwara, Punjab, India-144411.Journal Article20241123The goal of the current study is to offer a novel method for numerically solving coupled 1D and 2D nonlinear Schrödinger equations. To discretize the spatial partial derivative, we applied the MCNUAH B-spline DQM. The SSP-RK43 technique is used to solve the reduced system of ODEs. Via the matrix method, the stability of the proposed method is investigated, and it is found to be stable. Four experiments are used to confirm the efficiency of the suggested scheme, and data from the literature are compared throughout. It is clear that the obtained results are satisfactory and in strong accord with preceding results. This approach yields superior outcomes and is effective, straightforward, and reasonably simple to use. The graphical abstract is provided as per Figure 1.https://cmde.tabrizu.ac.ir/article_19400_2b628ed0df022df6142134d2a2f8c851.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A monotonic weighted compact finite difference solution for a non-linear steady advection-diffusion equation5495661907910.22034/cmde.2024.61722.2680ENPrapolChivapornthipDepartment of Industrial Engineering, Faculty of Engineering, Kasetsart University, Thailand.Journal Article20240520This paper introduces a monotonic weighted compact finite difference scheme (WC-FDM) designed to solve the nonlinear one-dimensional steady advection-diffusion equation (ADE). The WC-FDM scheme is validated against the analytical solution and is adaptable to accommodate both uniform and non-uniform grid spacing. Criteria for selecting weights have been developed to ensure scheme monotonicity. Computational performance is benchmarked against other numerical schemes. Numerical analyses reveal that the WC-FDM accurately solves the non-linear steady ADE for both uniform and non-uniform grid spacing scenarios without introducing spurious oscillations. The proposed weight criteria maintain the monotonicity of the WC-FDM scheme resulting the computational stability regardless of the advection dominance level and grid spacing uniformity.https://cmde.tabrizu.ac.ir/article_19079_96ef5a63bf8ac128cdc882208a65e0d5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A novel approach to fractional kinetic equations involving Srivastava polynomial and multi-index Bessel function5675801907110.22034/cmde.2024.61477.2651ENAlokBhargavaDepartment of Mathematics, Manipal University Jaipur, Jaipur, India.DayalalSuthar1. Department of Mathematics, Wollo University, P.O. Box 1145, Dessie, Ethiopia. \\ 2.Department of Mathematics, Saveetha School of Engineering, Thandalam 600124, Chennai India.Komal PrasadSharmaDepartment of Mathematics, NIMS University Rajasthan, Jaipur, India.Journal Article20240501In the present work, the generalized fractional kinetic equations (FKE) incorporating the composition of Multi<br />Index Bessel function and Srivastava polynomial are expressed with their fractional derivatives. Moreover, by<br />employing the idea of the Laplace transform, solutions are obtained in terms of the Mittag-Leffler function.<br />Finally, a numerical and graphical interpretation of the outcome is displayed.https://cmde.tabrizu.ac.ir/article_19071_2944c2e0f2464c5a6b605d267c007631.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Impact of fractional order on reaction rates: Solutions to kinetic equations with incomplete $\aleph$-function5815891933910.22034/cmde.2025.61995.2704ENShyamsunder-Department of Mathematics, SRM University Delhi-NCR, Sonepat-131029, Haryana, India.DikshaGangwarDepartment of Applied Mathematics, M. J. P. Rohilkhand University, Bareilly-243006, Uttar Pradesh, India.Journal Article20240607In this study, we investigate the significance of fractional kinetic equations in emerging a wide range of problems in science and engineering. Specifically, we derive a fractional kinetic equation solution involving the incomplete $\aleph$-function using a well-established integral transform technique. To illustrate the impact of the fractional integral operator’s order on reaction rates, we present several graphical results, highlighting the influence of fractional calculus on the system’s dynamics.https://cmde.tabrizu.ac.ir/article_19339_76ee0d7acd6b46ae04ee7e9a0f59c482.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Analyzing the MHD boundary layer flow of Rivlin-Ericksen fluid over a stretching sheet by applying the Taylor wavelet approach5906051919110.22034/cmde.2024.61873.2695ENPrithviSureshDepartment of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.Vidya ShreeRamareddyDepartment of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.Patil MallikarjunBasavarajDepartment of Studies and Research in Mathematics, Tumkur University, Tumkur-572103, Karnataka, India.Journal Article20240529In the current paper, we newly established the Taylor wavelet operational matrix method to study Revlin-Ericksen fluid flowing over the stretching sheet in the context of a magnetic field. The Taylor wavelet operational matrix method is a newly devised method for transforming the nonlinear differential equations to nonlinear algebraic equations. This computation is flexible and facile due to the generation of integral matrices. From these integral matrices, unresolved Taylor wavelet coefficients are determined with the help of solvers. Thus, the solution to the given Revlin-Ericksen fluid flow is achieved. This analysis examines the MHD Rivlin-Ericksen fluid flowing in the steady state caused by stretching a sheet while accounting for the inverse Darcy model. The aforementioned computational method is for seeking solutions to ordinary differential equations. Firstly, the momentum equation is changed to an ordinary differential equation by employing the similarity transformation, and then Taylor wavelet method has to be implemented for further analysis. The effect of the viscoelastic parameter, inverse Darcy number, magnetic parameter, and inclination angle on axial and transverse velocity are taken into consideration for study analysis. Engineering application tool local skin friction coefficient variation has been assessed for different parameters, and the estimated local skin friction coefficient is compared with bvp4c, demonstrating the compatibility of the Taylor wavelet approach.https://cmde.tabrizu.ac.ir/article_19191_f25a26235bd03eddaf4b3fd80c0cc963.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401An heuristic method for solving an inverse problem in semiconductors governed by a nonlinear coupled system6066151919810.22034/cmde.2025.62043.2708ENYounessEl YazidiResearch Laboratory in Numerical Analysis and Nonlinear Analysis (LaR2A) Department of Mathematics, Faculty of Sciences, University Abdelmalek Essaadi, BP 2121 M'Hannech II 93030 Tetouan, Morocco.AbdellatifEllabibLaboratory of Applied Mathematics and Computer Science, Faculty of Science and Technology, Cady Ayyad University, Marrakesh, Morocco.Journal Article20240610In this work, we utilize a coupled system to describe the carrier density in the Metal-Semiconductor Field-Effect Transistor (MESFET) device. To identify the depletion layer in this semiconductor, we define a cost functional $J$, and then use it to derive the shape optimization problem, for which we prove the existence of a solution. We develop an approach to solve this optimization problem using the finite element method combined with particle swarm optimization. Finally, we present several numerical examples to demonstrate the robustness of our proposed algorithm in identifying the depletion layer in the MESFET device.https://cmde.tabrizu.ac.ir/article_19198_15b6710a02c8f64ecb7c03166a73ef13.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Mathematical modelling with optimal control of infectious diseases with vaccination6166321946910.22034/cmde.2024.62350.2743ENHenry M.WanjalaChuka University, Chuka, Kenya.Mark O.OkongoChuka University, Chuka, Kenya.Jimrise O.OchwachMama Ngina University College, Gatundu, Kenya.Journal Article20240704Mathematical models are critical in provision of information to the development of infections. Not with standing the effectiveness of vaccines, some vaccinated individuals nonetheless get infected. To deal with this, non-pharmaceutical measures inclusive of social distancing and handwashing are encouraged. This study offers a mathematical version that combines the effects of vaccination and social distancing, utilizing Kermack-McKendrick compartments and ordinary differential equations (ODE's). The study determines the basic reproduction number ($R_0$) by the use of the Next Generation Matrix (NGM). If $R_0$ is less than 1, the ailment will in the end die out; if $R_0$ is more than 1, the ailment will continue to spread. Python simulations show that while vaccination and social distancing can reduce transmission, they may not be sufficient to eliminate the disease entirely. Isolation is critical for reducing transmission similarly, the efficacy of vaccines and the vaccination rate are crucial additives of a vaccination strategy. These techniques provide extra time for public health officers to put in force further measures, supplementing current processes. As we continue to come upon with new and evolving health challenges, the mixing of most reliable management strategies into epidemic modelling may be important. Further studies and interdisciplinary collaboration will enhance our capability to combat infectious sicknesses.https://cmde.tabrizu.ac.ir/article_19469_9ffc84c2254b11dc55a38cb1a99aab96.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401New optimal adaptive stepsize algorithm for solving black-scholes equation6336511972010.22034/cmde.2025.65090.2974ENMarziyehAlishahiDepartment of Mathematics and Computer Science, Lorestan University, Khorramabad, Lorestan 44316-68151, Iran.MajidYarahmadiDepartment of Mathematics and Computer Science, Lorestan University, Khorramabad, Lorestan 44316-68151, Iran.0000-0003-1286-7464Journal Article20241219In this paper, a new algorithm is designed based on a state feedback global error control system, Laplace trans form, order reduction method, and k-step numerical integration methods to numerically solve the Black-Scholes equation. For this purpose, the Black-Scholes equation is converted into a first-order system of ordinary differ ential equations by using the Laplace transform and order reduction method. Also, a new robust linear optimal adaptive global error control dynamic for designing an adaptive time variant step size sequence is modeled and a corresponding optimal control law based on robust and optimal eigenvalue assignment is designed. The proposed optimal control law guarantees the absolute stability of the implemented k-step numerical integrator methods. Finally, the transformed approximate solution of the Black-Scholes equation has been obtained using the Stefhest inverse Laplace transformation algorithm. The simulation examples show that the optimal control of global error under a given tolerance level, the guarantee of absolute stability, and the best approximation of sensitivity analysis indexes for the proposed approximate solution of the Black-Scholes equation is among the important advantages of the proposed method.https://cmde.tabrizu.ac.ir/article_19720_2d2d70b46ab14ad0f46f62806db4922d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401The behaviour of solutions boundary problem to nonlinear elliptic equations6526601927710.22034/cmde.2024.61634.2678ENTahirGadjievInstitute of Mathematics and Mechanics of NAS of Azerbaijan, The Azerbaijan Architecture and Construction University, Baku, Azerbaijan.Sardar YahyaAliyevBaku State University, Baku, Azerbaijan.Konul N.YasinliNakhchivan State University, Nakhchivan, Azerbaijan.Journal Article20240514We obtained estimates in generalized Morrey spaces are used to study global regularity of the solution of the Neumann boundary problem of nonlinear elliptic equations in divergence form over a bounded non-smooth domain. For these is we used of Calderon-Zygmund theory. The investigation of nonlinear reaction, drift, diffusion processes has received many attention. This questions arise as mathematical models of different applied problems. For example, for instance diffusion processes of electrically charged species phase transition and transport in porons media. The main goal of the article is to obtain Holder estimates for solutions to the Neumann problem for nonlinear elliptic equations.https://cmde.tabrizu.ac.ir/article_19277_ddcd8dad8d9b4f4cfd98f29c2dd0801a.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Jump-diffusion optimization: An iterative solution to the HJB equation for investment value6616772078110.22034/cmde.2025.68581.3328ENMehranPazireshFaculty of Mathematics, Statistics, and Computer Sciences, University of Tabriz, Tabriz, Iran.KarimIvazFaculty of Mathematics, Statistics, and Computer Sciences, University of Tabriz, Tabriz, Iran.Journal Article20250811This paper focuses on optimizing the investment value function by incorporating jump risk using the Merton Jump Diffusion (MJD) model. Our main goal is to determine the optimal dynamic asset allocation strategy to maximize expected utility. We derive the governing nonlinear Hamilton-Jacobi-Bellman (HJB) equation and employ a linearized generalized Newton method, which generates an iterative sequence for the optimal control. The theoretical convergence of this sequence was rigorously established using the Contraction Mapping Theorem, confirming the method's strong stability and reliability. Applying the model to real Google stock data, which exhibit significant jump risks, we derived an optimal investment ratio ($\mathbf{\pi^*}$) that suggests a notably aggressive allocation to the risky asset. This optimal strategy provides a direct, actionable benchmark for investors. Crucially, the derived dynamic control law functions as a powerful tool for investment management firms, enabling them to proactively adjust capital allocation strategies in response to potential future jump risk scenarios.https://cmde.tabrizu.ac.ir/article_20781_e577d8e5ff2aa33de7040d3c82d6643b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Hopf bifurcation and chaotic attractors in two special jerk system cases6786891914110.22034/cmde.2024.61865.2694ENTahsin IRasulDepartment of Mathematics, Faculty of Science, Soran University, Soran, Kurdistan Region, Iraq.Rizgar H.SalihDepartment of Mathematics, College of Basic Education, University of Raparin, Rania, Kurdistan Region, Iraq.Journal Article20240529This paper investigates the Hopf bifurcation with self-excited and hidden chaotic attractors in special types of chaotic jerk systems. The stability of the equilibrium points and Hopf bifurcation are rigorously analyzed for the proposed systems. It is remarkable to analyze the Hopf bifurcation using focus quantity techniques. These bifurcations can be either supercritical or subcritical, depending on the control parameters. The dynamic behavior of the systems, an analysis of self-excited chaotic attractors and hidden chaotic attractors was performed. Additionally, bifurcation analysis and evaluation of Lyapunov exponents revealed complex transitions among periodic, self-excited chaotic and hidden chaotic attractors as the system parameters varied. It was found that the systems exhibit both self-excited and hidden attractors, as demonstrated by the bifurcation diagrams, Lyapunov exponents and cross sections. All of the results provided in this study were acquired applying the Maple and Matlab software.https://cmde.tabrizu.ac.ir/article_19141_0d4a8898714d64a6c9d0c2f07b10128d.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Optical solitons and stability analysis for the improved Eckhaus equations6907001924510.22034/cmde.2024.62793.2784ENHinaZulfiqarDepartment of Mathematics, Government Sadiq College Women University, Bahawalpur, Punjab 63100, Pakistan.Kalim U.Tariq1. Department of Mathematics, Mirpur University of Science and Technology, Mirpur-10250 (AJK), Pakistan. \\ 2.Research Center of Applied Mathematics, Khazar University, Baku, AZ1096, Azerbaijan.Hamood UrRehmanDepartment of Mathematics, University of Okara, Okara, Punjab 56300, Pakistan.TaibaKouserApplied Research center of metrology and standard testing, King Fhad University of Petroleum and minerals Dhahran Saudi Arabia.WaqarAhmadDepartment of Mathematics, University of Okara, Okara, Punjab 56300, Pakistan.Journal Article20240804In this article, the propagation of modulated waves in one and two dimensional systems are analyzed by investigating the improved Eckhaus models analytically. Along with additional dimensions, dissipative factors, nonlocal effects, and higher-order nonlinear elements, the enhanced Eckhaus equation expands the original Eckhaus equation. The investigation of the governing models’ optical soliton solutions, including periodic, dark, brilliant, and singular solitons, is the focus of this article. This is done by obtaining a novel optical solution using the tanh-coth approach. Another type that incorporates nonlinearity and modulation effects in both spatial dimensions, and includes an extra spatial dimension, is the (2 + 1)-dimensional enhanced Eckhaus model. These equations are effective resources for examining a wide range of one- and two-dimensional system physical phenomena, including pattern generation, wave interaction, and soliton dynamics. Analyzing these equations can be challenging due to their higher dimensionality and nonlinear nature and numerical methods are often used to obtain solutions for specific cases or conditions. Consequently, trigonometric function solutions, hyperbolic function output and exponential functions solution with Independent parameters are acquired.3D and 2D contour plots of some solutions of the nonlinear model are specified. These governing equations have some applications in domains like nonlinear optics, condensed matter physics and fluid dynamics.https://cmde.tabrizu.ac.ir/article_19245_ed1bc44e221162e8231f2a41d184181b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A mathematical study on infinite boundary value problem for MHD flow of a micropolar nanofluid7017201934910.22034/cmde.2025.55531.2311ENVembuAnanthaswamyResearch Centre and PG Department of Mathematics, The Madura College (Affiliated to Madurai Kamaraj University), Madurai, Tamil Nadu, India.SakthivadeivelPunithaResearch Centre and PG Department of Mathematics, The Madura College (Affiliated to Madurai Kamaraj University), Madurai, Tamil Nadu, India.Journal Article20230222In this paper, the modified q-Homotopy analysis method (q-HAM) is employed to study the problem of magne tohydrodynamic (MHD) flow of nanofluid under buoyancy effects semi-analytically. The approximate analytic expressions of dimensionless velocity, dimensionless angular velocity, dimensionless temperature and dimension less concentration profiles are given explicitly. We can also derive the approximate analytical expressions for skin friction coefficient, Nusselt Number, and sherwood number. The graphical representation for numerous physical factors involved in the model are provided. This method is also extended to resolve various nonlinear problems in the applied sciences.https://cmde.tabrizu.ac.ir/article_19349_dd21498acf8c40be8ccafe275af9da06.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A highly accurate numerical technique for solving variable-order fractional Burgers-Huxley equation7217331970610.22034/cmde.2025.63198.2820ENMojtabaHajipourDepartment of Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran.YasharLakpourDepartment of Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran.Journal Article20240829In this article, we present a highly accurate technique for the numerical solution of the variable-order time-fractional Burgers-Huxley equation. The original equation is first discretized in the temporal and spatial directions. The third-order weighted-shifted Gr\"unwald-Letnikov and the fourth-order compact finite difference methods are used. We then formulate a nonlinear system of algebraic equations using the fully discretized version of the problem. The derived nonlinear system is solved by utilizing an iterative algorithm. The analysis of solvability, stability, and convergence of the method is also addressed. The method achieves a convergence rate of four in the spatial direction and three in the temporal direction. Moreover, it is a low-cost computational method and easy to implement. Finally, various illustrative examples are solved to verify the accuracy of the proposed method.https://cmde.tabrizu.ac.ir/article_19706_5154d81471d30519caafa89702ff2fa5.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401An advanced numerical approach for solving stiff initial value problems using a self-starting two-stage composite block scheme7347531975310.22034/cmde.2025.65372.3001ENBadrul AminJaafar1. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.\\
2. Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Sarawak, Kampus Samarahan 2, 94300 Kota Samarahan, Sarawak, Malaysia.Iskandar ShahMohd ZawawiFaculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia.Journal Article20250108In this study, a two-point composite block method based on the backward differentiation formula (CBBDF) is introduced to solve stiff ordinary differential equations. The CBBDF method incorporates an additional intermediate point among the interpolating points, developed in two stages: the first stage employs the Euler's method as a fundamental building block, while the second stage utilizes CBBDF of order three. A key distinction of the method with the classical block method is the introduction of an independent parameter $\gamma$, which eliminates the need for an external startup calculation, while maintaining the accuracy and stability of numerical solutions. The theoretical analysis verifies that the proposed method is convergent and $A$-stable. It fulfills the essential properties of consistency and zero-stability, and it lies within the $A$-stability region. To demonstrate the effectiveness of the proposed approach, several stiff initial value problems of linear and non-linear are solved. For validation, the results are compared with existing literature. While approximating the solution at multiple points simultaneously, the composite block method offers the ability to use larger step sizes for solution approximation. The CBBDF method shows promising results, achieving a reliable degree of accuracy as indicated by its maximum error and average error measurements.https://cmde.tabrizu.ac.ir/article_19753_56f1148829b21bd6ba0c8bf61493943b.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Infinitely many conservation laws, multi-wave solutions, and interactions for the (2+1)-dimensional complex modified Korteweg-de Vries system of equations7547651940110.22034/cmde.2024.62458.2753ENThilagarajahMathanaranjanDepartment of Mathematics and Statistics, University of Jaffna, Sri Lanka.Journal Article20240712In this paper, we consider the (2+1)-dimensional complex modified Korteweg-de Vries (cmKdV) system of equations. This system of equations is a generalization of the cmKdV equation in the (2+1)-dimension and has great significance in the fields of applied magnetism and nanophysics. On the basis of the Lax pair, infinitely many conservation laws are obtained. In addition, the multi-waves, homoclinic breather, rational, and interactions solutions of this equation are derived with the aid of logarithmic transformation and symbolic computation. For the suitable value of parameters, the 3D surfaces of obtained solutions have been plotted using Mathematica.https://cmde.tabrizu.ac.ir/article_19401_8990e049388034ffe087fd391750ae43.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401$L^{\ell}-$Asymptotic properties of nonlinear Sturm-Liouville problems7667791971810.22034/cmde.2025.64894.2957ENFatemehKiyaeeDepartment of Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.SeyfollahMosazadehDepartment of Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran.Journal Article20241206In this paper, a nonlinear eigenvalue problem consisting of a nonlinear Sturm-Liouville equation $-y''- q(x)y= \lambda q^{-1}(x) y^{r}$ with Dirichlet boundary conditions on the interval $(-1/2 , 1/2)$ is investigated, where $\lambda >0$ is the eigenparameter. We provide a simple scheme to obtain the asymptotic behavior of $L^{\ell}-$bifurcation curve $\lambda=\lambda_{\ell}(\gamma)$ as $\gamma\longrightarrow 0$, where $\gamma=|| y_{\lambda}||_{\ell}$, $\ell \geq 1$, and $y_{\lambda}$ is the solution of Dirichlet problem associated with $\lambda$.https://cmde.tabrizu.ac.ir/article_19718_9cf090a0529b5a173416546ba892bbb1.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401New explicit solution of the Blasius equation in the boundary layer around the hull of a ship by approximation of derivatives7807971937010.22034/cmde.2024.62192.2724ENAbdelkaderLahlaliLaboratoire de Mécanique d’Ingénierie et d’Innovation LM2I, ENSEM, Université Hassan II, Casablanca 20103, Morocco.ZakariaEl MaskaouiLaboratoire de Mécanique d’Ingénierie et d’Innovation LM2I, ENSEM, Université Hassan II, Casablanca 20103, Morocco.LabhbiBoussihineLaboratoire de Mécanique d’Ingénierie et d’Innovation LM2I, ENSEM, Université Hassan II, Casablanca 20103, Morocco.BouchabNadirDépartement Energie, Ecole Royale Navale, Casablanca 20052, Morocco.AbderrahimDinaneDépartement Energie, Ecole Royale Navale, Casablanca 20052, Morocco.Journal Article20240620Naval hydrodynamics fundamentally depends on a detailed understanding of the boundary layers forming around a ship’s hull, which generate resistance to advancement. Accurately modeling these layers is critical for calculating hydrodynamic resistance and estimating the propulsion power needed to achieve the desired speed specified by the shipowner. Traditionally, the velocity distribution within the boundary layer is described by the Blasius equation, a nonlinear third-order differential equation commonly solved using the Runge-Kutta numerical method, renowned for its accuracy.<br />This study proposes a novel direct and explicit approach to solving the Blasius equation around a ship’s hull, leveraging a derivative approximation technique implemented with MATLAB to obtain numerical results. By employing sufficiently small step sizes, the method produces highly accurate results that can serve as a benchmark for evaluating the precision of other numerical techniques applied in ship design. The proposed derivative approximation method provides a simple yet robust tool for solving complex differential equations, demonstrating its potential as an effective alternative for tackling problems similar to the Blasius equation in naval engineering applications.https://cmde.tabrizu.ac.ir/article_19370_44cc0b40905f6bef77f35ec0f6b2e33e.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401An overlapping adaptive step-size multi-derivative hybrid block method for higher order initial value problems7988271962410.22034/cmde.2025.63158.2817ENUthman OlamideRufaiSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg 3201, South Africa.PreciousSibandaSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg 3201, South Africa.SiceloGoqoSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Pietermaritzburg 3201, South Africa.Journal Article20240828The need for accurate solutions to mathematical models, particularly for linear and nonlinear higher-order initial value problems, is essential across various scientific and engineering fields. Traditional methods often face challenges with stability and precision, especially in non-linear cases, prompting the development of advanced numerical techniques. This study introduces a two-step overlapping adaptive step-size multi-derivative hybrid block method to address these challenges in solving higher-order initial value problems. The method incorporates overlapping elements, using the second-to-last intra-step point from the previous step within each integration block to enhance accuracy. The method uses error estimation and selects an appropriate step-size, ensuring the desired accuracy without wasting computational resources or introducing unnecessary errors. The non-linear initial value problems are efficiently linearized using a modified-Picard iteration. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed method, and its performance is compared against a similar non-overlapping method as well as other methods reported in the literature.https://cmde.tabrizu.ac.ir/article_19624_346487b5e1d9a53fa52153509062b2b2.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A study of a fractal multi-pantograph delay model with varying coefficients using fractional order wavelets8288511970310.22034/cmde.2025.62311.2735ENDeepakSinghDepartment of Mathematics and Statistics, Gurukula Kangari (Deemed to be University), Haridwar, 249404, Uttarakhand, India.Sag RamVermaDepartment of Mathematics and Statistics, Gurukula Kangari (Deemed to be University), Haridwar, 249404, Uttarakhand, India.Journal Article20240701Apart from using fractal dimensions to describe statistical self-similarity, exploring a direct measurement approach involves considering mathematical models capable of constructing a real-world fractal entity, as classical differential and integral operators cannot efficiently handle such problems. In this study, the fractal derivative is applied to develop a fractal model for multi-pantograph delay differential equations with variable coefficients. The wavelet approach, employing Jacobi fractional order wavelets, has been developed to attain a numerical solution. The proposed methodology relies on the utilization of the fractal integral operational matrix of Jacobi fractional-order wavelets combined with the collocation method. We have outlined pseudo-code and conducted a stability analysis for the methods proposed within the specified model. Furthermore, the convergence analysis of the approximate solution is presented through some lemmas and theorems. The numerical results and error analysis of some illustrative examples are also shown in the tables and graphs. In the proposed methods, numerical results are derived across various values of the fractal $(\mu)$ and fractional $(\phi)$ parameters. It is important to highlight that the classical scenario is retrieved when $\mu=1$.https://cmde.tabrizu.ac.ir/article_19703_a1228dab8228d96a4fe9834c5ef885e4.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401A computational iterative technique for a kind of nonlinear higher-order singular Emden-Fowler type equations8528681972110.22034/cmde.2025.65131.2979ENJyoti-Department of Mathematics, JUIT Solan, Waknaghat-173234, Himachal Pradesh, India.MandeepSinghDepartment of Mathematics, JUIT Solan, Waknaghat-173234, Himachal Pradesh, India.Journal Article20241223This paper examines the solutions of nonlinear higher-order singular Emden-Fowler type equations arising in various physical models. Generally, it becomes difficult to obtain the solution near the point of singularity. To overcome this problem, an iterative technique is introduced that depends on the variational iteration method (VIM) and the homotopy perturbation method (HPM). Such a technique generates the solution in terms of a series, which is highly practical from computing perspective. An equivalent recursive integral representation (involving Lagrange's multiplier) for the higher order nonlinear singular Emden-Fowler type (SEFT) equations with initial conditions (ICs) is established with the support of the variational iteration method (VIM). Making use of the concept of homotopy, a system of integral equations is established, which helps to deal with nonlinearity. Some numerical examples are studied through the proposed iterative technique to show the applicability and efficiency of the technique.https://cmde.tabrizu.ac.ir/article_19721_8d4bb0e2ddf885564678895f69e0c293.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Power series solutions of fractional Lotka-Volterra equation8698761919910.22034/cmde.2025.61524.2662ENJichengYuSchool of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, China.YuqiangFeng1. School of Science, Wuhan University of Science and Technology, Wuhan 430081, Hubei, China.
2. Hubei Province Key Laboratory of Systems Science in
Metallurgical Process, Wuhan 430081, Hubei, China.Journal Article20240505In this paper, the power series method is applied to the fractional Lotka-Volterra equation, one of the most famous competition models in demography and economics. We obtain some power series solutions of the governing equation and prove their convergence. In addition, we analyze the various types of competitive roles depicted by this model through the truncated graphs of these power series solutions. From the graphs, we can find that the fractional order affects the speed of population growth or decrease, and this effect can be seen as continuous with respect to the order.https://cmde.tabrizu.ac.ir/article_19199_d17fa277f567132ddaef57520cfec64c.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Inverse optimization problem for a fractional analog of the Barenblatt--Zheltov--Kochina equation8779071928110.22034/cmde.2025.62338.2740ENTursun K.Yuldashev1. Tashkent State Transport University, Temiryulchilar Street 1, Tashkent, 100167 Uzbekistan.
2. Alfraganus University, Yukori Karakamysh street 2A, Tashkent, 100190 Uzbekistan.AyselRamazanovaUniversität Duisburg-Essen, Thea-Leymann-Straße 9, D-45127 Essen, Germany.Journal Article20240703The generalized solvability of a nonlinear optimal control for thermal and diffusion processes in a mixed inverse problem for a Barenblatt-Zheltov-Kochina differential equation with Hilfer fractional operator is studied. The inverse problem is considered with spectral and intermediate conditions. Eigenvalues, eigenfunctions, and associated functions of the spectral problem are found and the corresponding adjoint problem is solved. Countable systems of fractional order differential equations with final value conditions are obtained. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to solve a complicated nonlinear functional integral equation, and the process of solving consists of solving separately taken two nonlinear functional-integral equations. Nonlinear functional integral equations are solved by the method of successive approximations and the unique solvability of these equations is proved by the method of contracting mapping. Approximate calculations for the optimal control function, the redefinition function, and the state function of the controlled process are obtained. The absolute and uniform convergence of the obtained Fourier series are proved.https://cmde.tabrizu.ac.ir/article_19281_e3891eba32cbfda4ba49fe459e2535bd.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Stability analysis of SAIR mathematical model with general incidence rates and temporary immunity9089181939410.22034/cmde.2025.64267.2901ENNematNyamoradiDepartment of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran.KaramAllaliLaboratory of Mathematics, Computer Science and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, PO Box 146, Mohammedia, Morocco.BashirAhmadNonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia.Journal Article20241030This paper studies the dynamics of a SAIR mathematical model that describes the interaction among susceptible, asymptomatic, symptomatic, and recovered individuals. Two general incidence functions describing the infection caused by the asymptomatic and symptomatic individuals are introduced. We also take into account a temporary immunity, that is, a proportion of the recovered individuals becomes susceptible again. The basic reproduction number $R_0$ depends on the general incidence functions. The local and global asymptotical stability for each equilibrium will depend on the basic reproduction number $R_0$. In precise terms, the disease-free equilibrium is locally and globally asymptotically stable when $R_0<1$, while the endemic equilibrium is locally and globally asymptotic stable when $R_0>1$. The numerical simulation is performed for different incidence rate cases, such as bilinear, Beddington-DeAngelis, Crowley Martin, and non-monotonic incidence rate functions. The simulation results are found to agree with the theoretical endings.https://cmde.tabrizu.ac.ir/article_19394_3708a402d5866e690c71a6c492eadc41.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Multi-soliton solutions to the K-P equation of tenth-order9199281917210.22034/cmde.2024.56556.2366ENBharathaKalegowdaPG Department of Mathematics, PG Studies and Research Centre, St. Philomena's College, Mysuru-570 015, India.RangarajanRaghavacharDepartment of Studies in Mathematics, University of Mysore, Manasagangothri, Mysuru-570 006, India.Journal Article20230511Kadomtsev–Petviashvili (KP) equation is an important (2+1) - dimensional nonlinear PDE which has not only multi-solitons but also has complete integrability. In order to describe the long waves that propagation weakly dispersive in the direction of additional spatial variable $y$, Kadomstev and Petviashili formulated this model. In the literature, many researchers are interested to propose and work on higher order nonlinear PDEs possessing multi-solitons. Two powerful methods employed by researchers are Hirota's method to obtain multi-solitons and $\tanh-\coth$ method to obtain single-soliton solutions. In our work, a tenth-order generalization of the KP equation is derived and using Hirota's method, its multi-solitons are worked out. Furthermore, the derived equation is also treated with the $\tanh$ method. This article emphasizes few bounded solutions to the equation in context. The main aim of this paper is to demonstrate the generalization of the K-P equation using Hirota operators and to study corresponding multi-solitons. Finally, some open problems related to the proposed tenth-order KP equation are discussed.https://cmde.tabrizu.ac.ir/article_19172_976dbb8d6b498a680586ca74043a5d66.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Soliton solutions in the nonlinear conformable Wu-Zhang system9299461970110.22034/cmde.2024.61514.2661ENHiraTariqDepartment of Mathematics, Government College Women University, Sialkot, Pakistan.HiraAshrafDepartment of Mathematics, Government College Women University, Sialkot, Pakistan.Mohammad AliHosseinzadehFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, Iran.HadiRezazadehFaculty of Engineering Modern Technologies, Amol University of Special Modern Technologies, Amol, Iran.SoheilaBiriaHigher Education Extension Office, Vice-Chancellor of Education, Ministry of Science, Research and Technology, Tehran, Iran.Journal Article20240504In this paper, new analytical solutions of nonlinear fractional Wu-Zhang system are determined with the aid of two analytical approaches, that is, generalized projective Riccati equation method and Sardar sub-equation method via conformable derivative. The system describes (1 + 1)-dimensional dispersive long wave in two horizontal directions on shallow waters. Some new solitary wave solutions are demonstrated by the means of computer softwares maple or mathematica. The obtained results reveals that the proposed method is very efficacious and straightforward in the determination of the solution for the nonlinear fractional Wu-Zhang system.https://cmde.tabrizu.ac.ir/article_19701_1e3db1dd23304dc11dcf414b4566a275.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Implicit cubic B-spline quasi-interpolation for solving the generalized distributed-order time-fractional Black-Scholes equation9479571974910.22034/cmde.2025.64886.2956ENRoyaMontazeriDepartment of Mathematics, Payame Noor University (PNU), P.O.Box 19395-4697, Tehran, Iran.Journal Article20241206In this study, we implicitly solve the generalized distributed-order time-fractional Black-Scholes equation. We employ finite differences to approximate the time derivatives and cubic B-spline quasi-interpolation for the spatial derivatives. The error analysis of the presented method is investigated. The algorithm of this method is also presented, which shows the simplicity of implementing the method to solve the generalized distributed-order time-fractional Black-Scholes equation. Numerical results demonstrate the method’s convergence rate and accuracy.https://cmde.tabrizu.ac.ir/article_19749_f8ccf4d0bca74d38982f1098c89bb777.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Symmetry properties, exact solution and conservation laws of time-fractional Zeldovich-Frank-Kamenetskii equation9589681975710.22034/cmde.2025.64871.2954ENParastooKabi-NejadFaculty of Pure Mathematics, School of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran, Iran.Journal Article20241205The article focuses on investigating Lie symmetry analysis of the time-fractional Zeldovich-Frank-Kamenetskii equation with Riemann-Liouville derivative. The fractional reaction-diffusion equation describes how planar laminar premixed flames spread in combustion theory. The use of the Lie method is also illustrated to obtain Lie symmetry generators, symmetry reduction solutions, invariant properties, and conservation laws. Furthermore, we convert the time-fractional Zeldovich-Frank-Kamenetskii equation to a nonlinear fractional ordinary differential equation (ODE) with Erd\'{e}lyi-Kober derivative using its Lie point symmetries. This decreased fractional ODE is investigated by explicit power series. In addition, some figures for the obtained explicit solution are presented.https://cmde.tabrizu.ac.ir/article_19757_51878580b3bd34a14aeb50029940ba0f.pdfUniversity of TabrizComputational Methods for Differential Equations2345-398214220260401Bioconvective flow surrounding a thin surgical needle in blood incorporating ternary hybrid nanoparticles9699951891710.22034/cmde.2024.61425.2643ENAhmed S.Rashed1. Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig university, Egypt.
2. Basic Science Department, Faculty of Engineering, Delta University for Science and Technology, Gamasa, 11152, Egypt.EhsanNasrDelta Higher Institute for Engineering and Technology, Mansoura, Egypt.Samah M.MabroukDepartment of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig university, Egypt.Journal Article20240429Biological systems use fluid dynamics to coordinate group movements and spatial arrangement, which affect both their own dispersion and the dynamics of their surroundings. This behavior has been documented in a number of biological systems, such as bacterial colonies, algal blooms, and microbial suspensions. The current study examines the flow of a nanofluid via a vertical thin needle used in medical surgery. The nanofluid is composed of three types of nanoparticles: Fe3O4, copper oxide (CuO), and copper (Cu) that are dispersed in a base fluid of blood. Additionally, the nanofluid contains gyrotactic bacteria. Furthermore, in the presence of a magnetic field, the incompressible liquid conducts current. The nanofluid model considers both Brownian motion and thermophoresis. The Runge-Kutta and shooting approach is used to numerically solve transformed ODEs resulting from the group method. The present study looked at the effects of several factors, including Prandtl number, Brownian motion coefficient, thermophoresis diffusion coefficient, microorganism diffusion coefficient, concentration difference, temperature difference, Schmidt number, bioconvection Peclet number, Lewis number, and magnetic diffusivity. The findings indicate that velocity decreases with rising \(\Pr,Lb\ \) and \(Sc\) and increases with \(D_{B}\), \(D_{T}\), \(D_{n}\), \(\delta c\), \(\delta t\), and \(Pe\). In contrast, temperature decreases with increasing \(\Pr\), \(D_{B}\), and \(\delta c\) and increases with rising \(\delta t\). Bacterial density, on the other hand, decreases with rising \(\Pr\) and \(D_{B}\) and increases with \(D_{T}\), \(D_{n}\), and \(Sc\). Whereas the magnetic field grows as \(\eta_{0}\) increases. We will also use graphs to illustrate the physical significance of the current parameters.https://cmde.tabrizu.ac.ir/article_18917_f1c07d3940fb0b6cc178136f7530ba6f.pdf