This article presents a quintic B-spline method to find an approximate solution of the second-order singularly perturbed differential equation in which the convection term occurs with a negative shift. The proposed method gives rise to a pentadiagonal linear system. Thomas' algorithm is employed to solve the obtained system of equations. The method's convergence is examined through truncation error analysis, and the existence and uniqueness of the solution are also established. Maximum absolute error is tabulated for two numerical examples, proving the proposed method's efficiency. Graphs are drawn to show the behavior of the solution. A comparative study shows that the obtained solution is better than the previous solutions in the literature. The method is found to be fourth-order convergent. The effect of the delay parameter on the boundary region is also discussed in the example.

The Newell-Whitehead-Segel (NWS) equation plays a significant role in nonlinear systems, including mathematical biology, plasma physics, solid-state physics, optics, quantum mechanics, cosmology, fluid dynamics, and many others. In this work, we proposed the quintic B-spline collocation method to find the numerical solution of the nonlinear NWS-type equation. Crank-Nicolson finite difference method (FDM) is used to discretize the equation in time space, and quasi-linearization is employed to linearize the nonlinear term. The stability analysis has been discussed using the Von Neumann Method, and stability conditions have been obtained. The numerical results are compared with existing techniques, which demonstrate the effectiveness and applicability of the proposed technique. The proposed method has been applied to four numerical test problems at various time levels and mesh sizes to demonstrate the effectiveness, which involves quadratic, cubic, and quartic order nonlinear terms. The comparison shows good agreement with the exact solution, as demonstrated by absolute error tables and graphs. Moreover, the proposed method is easy to implement and produces good results.

The heat and mass transfer behavior of a convective magnetohydrodynamic (MHD) Maxwell nanofluid past a stretching cylinder with activation energy and chemical reaction effects is addressed in this article. The overall goal is to quantify the joint influences of viscoelasticity, magnetic forces, Brownian motion, thermophoresis, heat generation, Arrhenius activation energy, and chemical reactions, and nanoparticle transport mechanisms on momentum, thermal, and nanoparticle concentration distributions. The process of reducing the governing nonlinear equations of motion to ODE’s via similarity transformations and numerically solving them by the RKF45 method with a shooting method at a convergence tolerance of 10−6 has been detailed in this paper. The results indicate that as the Hartmann number increases, the Lorentz forces act to decelerate the velocity, but they contribute to an increase in both temperature and the nanoparticle concentration. The increasing Maxwell parameter represents an increase in fluid elasticity and causes a reduction in velocity. The enhanced thermal diffusion resulting from the Brownian motion will raise the temperature of the fluid, and the resultant enhanced thermophoretic motion will promote the thermal transfer of energy through the fluid. The study also reveals that increasing the thermal Biot number from 0.1 to 0.4 enhances the heat transfer rate by about 91.8 %, confirming the strong influence of surface convection on thermal transport. The chemical reaction parameter from 0.05 to 0.25 reduces nanoparticle mass transfer by nearly 11.1 %, indicating suppressed concentration flux due to intensified reaction effects. In contrast, activation energy from 1 to 4 significantly improves mass transfer rate, yielding an approximate 50.3 % increase in the Sherwood number as weaker effective reactions allow greater nanoparticle accumulation near the surface. The Nusselt number calculated in this study is in excellent agreement with previously published literature. The novelty of this work lies in being the first to incorporate a Maxwell fluid framework to model stretching alongside activation energy controlled chemical kinetics over cylindrical surfaces. Results of this study can be of interest to researchers studying polymer processing, fiber coating, thermal management systems, energy conversion devices, and controlled drug-delivery methods, where strict control of heat transfer and dispersion of nanoparticles is important.

In this paper, we developed an effective computational technique for addressing third-order linear singularly perturbed problems having the source term discontinuous. Boundary or interior layers are frequently present in singular perturbation issues, making traditional numerical techniques more challenging. Here, we present a quartic B-spline method (QBSM) for the approximate solution of the third-order singularly perturbed boundary value problem, improving both the accuracy and efficiency of the solutions. In addition, the proposed method's convergence and error are investigated. The performance of the current technique is demonstrated through numerous numerical tests. The numerical findings are compared to other approaches reported in the literature.

The Hunter-Saxton (HS) nonlinear partial differential equation, which is used to model the nematic liquid crystals and describes certain aspects of the orientation wave. In this study, we proposed the exponential cubic B-spline collocation method to solve the HS equation. The equation in space and time is discretized using the exponential cubic B-spline method and the Crank-Nicolson finite difference approach, respectively. The nonlinear components are linearized by quasi-linearization technique. Discretization techniques in both time and space are used over uniform meshes, which leads to a tridiagonal system. The Von Neumann method has been employed to analyze stability and to determine stability conditions. Two numerical examples have been examined at different values of t and mesh points to demonstrate the effectiveness of the proposed method. Error tables and graphs illustrate the results which are compared with the existing literature. The results demonstrate strong agreement with the analytical solution and outperform the trigonometric cubic B-spline collocation method.

This manuscript presents a Quartic B-spline Method (QBSM) for the accurate estimation of numerical solutions to singularly perturbed third order boundary value problems (SPTOBVPs). The technique leverages a piecewise uniform mesh, constructed through the Fitted Mesh Technique (FMT), to enhance nodal point distribution within boundary layer regions. The convergence of proposed method are thoroughly investigated and found that it gives second-order convergence results. The efficacy of proposed method is demonstrated through the numerical examples, substantiating its theoretical results.

Various physical phenomena give rise to singularly perturbed differential equations with mixed shifts. Due to multiple parameters, singularly perturbed mixed delay boundary value problems are challenging to solve. This article considers a singularly perturbed differential-difference equation with delay and advance. To deal with the complexity of these equations, a non-polynomial spline numerical approach is adopted. For discretization, we have used a uniform mesh with equal spacing. Various theoretical results like stability and convergence are discussed. Numerical examples are solved to support the method and check the validity of the findings. The numerical order of convergence is determined and presented in the tables, along with the comparison of the results with the other existing methods. The comparison shows that the error is significantly less than the available solution in the literature. Also, the numerical order of convergence is determined to be equal to two. Graphs are drawn to observe the behavior of the solution for different values of parameters.

This research presents an efficient and highly accurate cubic B-spline method (CBSM) for solving second-order linear boundary value problems (BVPs). The method achieves sixth-order convergence, supported by rigorous error analysis, ensuring rapid error reduction with mesh refinement. The effectiveness of the CBSM is validated through four numerical examples, showcasing its accuracy, reliability, and computational efficiency, making it well-suited for large-scale problems. A comparative analysis with existing methods confirms the superior performance of the CBSM, positioning it as a practical and powerful tool for solving second-order BVPs.

This research addresses the numerical solution of singularly perturbed convection-diffusion kind boundary value problem of second-order with a discontinuity term. Due to the perturbation parameter and discontinuity term, the problem solution has a boundary layer and an interior layer. A nonpolynomial cubic spline method is utilized to solve the boundary value problem. A specific set of parameters associated with nonpolynomial spline is used to tailor the method. A comprehensive analysis of the stability and convergence of the recommended method is presented which gives second-order convergence results. The suggested method is implemented on two examples, and the obtained results are contrasted with an existing method, highlighting the precision and efficacy of the proposed method, which would enhance the method's novelty.

This paper introduces a novel computational approach utilizing the quartic B-spline method on a uniform mesh for the numerical solution of non-linear singularly perturbed delay differential equations (NSP-DDE) of second-order with a small negative shift. These types of equations are encountered in various scientific and engineering disciplines, including biology, physics, and control theory. We are using quartic B-spline methods to solve NSP-DDE without linearizing the equation. Thus, the set of equations generated by the quartic B-spline technique is non-linear and the obtained equations are solved by Newton-Raphson method. The success of the approach is assessed by applying it to a numerical example for different values of perturbation and delay parameter parameters, the maximum absolute error (MAE) is obtained via the double mesh principle. The convergence rate of the proposed method is four. Obtained numerical results are compared with existing numerical techniques in literature and observe that the proposed method is superior with other numerical techniques. The quartic B-spline method provides the numerical solution at any point of the given interval. It is easy to implement on a computer and more efficient for handling second-order NSP-DDE.

The stretching surface flow phenomenon plays novel significance in the differential industrial sectors like manufacturing processes, food processing, polymers, petroleum engineering, glass fiber production, metal spinning, and many others. This research focuses on investigating the flow of electroosmotic-driven transport of magnetic Casson fluid over an exponential stretching sheet. The study employs mathematical modeling based on partial differential equations which are further truncated into ordinary system under the implementation of dimensionless variables. The quartic B-spline method is applied for obtaining solutions. Key findings indicate that electroosmotic and Helmholtz-Smoluchowski effects enhance fluid velocity, while magnetic forces reduce velocity. Additionally, an increase in the Hartmann parameter leads to a decrease in the skin friction coefficient.

This study investigates the peristaltic transport of Jeffrey nanofluid in a physiological vessel, addressing the significant issue of optimizing fluid transport in biomedical and industrial applications, such as targeted drug delivery, thermal management devices, and biosensor technologies. Approximate analytical solutions were derived using long wavelength and low Reynolds number approximations to simplify the complex system and provide clear insights into fluid dynamics. The study incorporates an applied magnetic field, viscous dissipation, heat sources, electroosmosis, and thermal radiation. Significant outcomes include higher temperatures in blood-graphene nanofluid compared to blood-platinum nanofluid, reduced nanofluid velocity with increased magnetic field strength, higher irreversibility generated by blade-shaped nanoparticles, and reduced bolus size with increasing Jeffrey fluid parameter. These findings highlight the complex interactions between various physical parameters and suggest optimization strategies for specific applications. The study’s goal is to provide a foundation for future research and practical implementations in relevant technologies.

A “Circular Supplier Selection (CSS)” is a procedure of choosing suppliers in a “Closed-Loop Supply Chain (CLSC)”. A “Sustainable Circular Supplier Selection (SCSS)” discusses the environmental and social issues in the region of “Circular Economy (CE)” and “Sustainable Supplier Selection (SSS)”. Due to the subjectivity of human judgment, the best SCSS often comprises ambiguous and indeterminate data, and the “Single-Valued Neutrosophic Sets (SVNSs)” have a vast proficiency to treat uncertain, indeterminate, and inaccurate information in the “Multi-Criteria Decision-Making (MCDM)” process. This chapter develops an extended methodology with the “Additive Ratio Assessment (ARAS)” and “Criteria Importance Through Intercriteria Correlation (CRITIC)” under SVNSs called SVN-CRITIC-ARAS method and utilizes it to treat the SCSS in the manufacturing companies. The comparison with extant models shows that the SVN-CRITIC-ARAS tool is capable and proficient to select the best sustainable circular supplier. Hence, the developed framework can be implemented by companies to evaluate and find the appropriate supplier in the SCSS procedure in the CE.

This study investigates the dynamics of couple stress ternary nanofluid flow over a stretching sheet by incorporating Al2O3, SiO2, and TiO2 nanoparticles into Sodium alginate, addressing the critical gap in understanding the behavior of such suspensions in thermal management applications. The importance of this research lies in its potential to enhance heat transfer efficiency in processes like polymer extrusion and cooling systems. The governing non-linear partial differential equations were transformed into ordinary differential equations using similarity transformations, facilitating detailed analysis of temperature distribution under prescribed surface temperature (PST) conditions. Key results indicate that increasing the magnetic field strength and the inverse Darcy number lead to a reduction in fluid momentum. Conversely, higher thermal radiation and heat source parameters significantly elevate the temperature profile. This work is novel as it integrates multiple nanoparticles within a single suspension and explores their effects on flow dynamics and thermal properties in the context of a stretching sheet, surpassing previous literature that primarily focused on binary or hybrid nanofluid systems.

In this article, we investigate a hybrid difference scheme for finding the numerical solution of a singularly perturbed second-order reaction-diffusion problem with a discontinuous source term.Such types of problems arise in the modeling of semiconductor devices and geophysical fluid dynamics etc. Solutions of these types of problems are difficult to obtain due to the presence of boundary and interior layers.A hybrid difference scheme i.e., cubic spline method and central finite difference approach, are applied on a fine region and coarse region, respectively.Shishkin mesh is utilized to generate the mesh point for the given domain.We use a second-order hybrid difference operator at the point of discontinuity.The solution rapidly changes in the interior layers and boundary layer.Truncation error is studied, and the stability of the method is analyzed.The proposed method is implemented on two problems, and numerical results are compared with the existing method, which shows that the proposed method is efficient for reducing maximum absolute errors and increasing the rate of convergence.

The current research proposes a novel efficient quintic B-spline (QBS) numerical technique based on piecewise uniform mesh for the numerical solution of fourth-order singularly perturbed boundary value problems (SPBVP) with discontinuous source terms (DST). The fifth-degree basis spline functions are employed along with a new approximation for the fourth-order derivative without altering the order of differential equations in which they appear. The proposed method is verified based on two problems to corroborate the scheme. The comparison of computational results shows that the proposed approach outperforms the existing approach in the literature. The stability and truncation error of the proposed way is analyzed.

In this article, we extended operational matrices using orthonormal Boubaker polynomials of Riemann-Liouville fractional integration and Caputo derivative to find numerical solution of multi-term fractional-order differential equations (FDE). The proposed method is utilized to convert FDE into a system of algebraic equations. The convergence of the method is proved. Examples are given to explain the simplicity, computational time and accuracy of the method.

Over the years, the squeezing fluid motion between parallel plates has generated the research interest in many investigators due to its various engineering and industrial applications such as injection modeling, compression and polymer process industry. With these motivations, in the current article, the transient state electro-osmotic squeezing propulsion of viscous liquid between two parallel plates through a porous medium with the impact of radiative heat flux has been examined. The applied magnetic field inclination angle ranges from 0 to 90 degrees and moving closer together causes the plates to squeeze one another orthogonal to the plates' surfaces. Through the suitable similarity transformations, the nonlinear partial differential equations characterizing the proposed flow model are reduced to a set of nonlinear ordinary differential equations. The numerical findings were achieved via MATLAB software. It is determined that the applied electric and magnetic field have the impact on the velocity and heat transfer of squeezing flows. Increasing the electroosmosis parameter as well as the electric field parameters have a considerable effect on temperature augmentation. The results of the current study can be used in various industrial applications such as chemical engineering, hydraulic lifts, polymer processing and power transmission.

This study presents a quintic B-spline collocation method (QBSCM) for finding the numerical solution of non-linear Bratu-type boundary value problems (BVPs). The error analysis of the QBSCM is studied, and it provides fourth-order convergence results. QBSCM is applied on two numerical examples to exhibit the proficiency and order of convergence. Obtain results of the QBSCM are compared with other existing methods available in the literature.

In this paper, we apply a quintic B-spline method for solving two-parameter second-order singularly perturbed boundary value problems. By using this method on piecewise uniform Shishkin mesh, we get a pentadiagonal linear system of equations. The convergence analysis has been established, and the method is shown to have uniform convergence of order four. Numerical results support the theoretical results. Relevance The work contained in this communication is mainly focused on the numerical solution of two-parameter singularly perturbed boundary value problems via quintic B-spline method. The proposed technique plays an important role to obtain the better numerical solution of these types of the problems and it is also used in image processing, computer graphics and surface fitting problems, etc.