Numerical analysis, Singular Perturbation Boundary Value Problems, Spline Methods
35
Scopus Publications
360
Scholar Citations
9
Scholar h-index
9
Scholar i10-index
Scopus Publications
Numerical Solution of Newell-Whitehead-Segel Equation via Quintic B-spline Collocation Method Nilesh H. Thube, Ram Kishun Lodhi International Journal of Mathematical Engineering and Management Sciences, 2026 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.
Numerical solution of Hunter-Saxton type equation via exponential cubic B-spline collocation method Nilesh H. Thube, Ram Kishun Lodhi Boundary Value Problems, 2025 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.
Leveraging DeepLabv3+ and U-Net Models for Accurate Spine Segmentation in CT Scans Retinderdeep Singh, Chander Prabha, Meena Malik, Ram Kishun Lodhi International Journal of Mathematical Engineering and Management Sciences, 2025 Medical imaging depends heavily on spine segmentation for diagnosing spinal disorders as well as developing treatment plans. The proposed research describes a deep learning framework for spine segmentation where two state-of-the-art models known as DeepLabv3+ and U-Net are evaluated for comparison. Our team applied the developed framework using 1089 high-resolution CT images which had been annotated with segment outlines for cervical, thoracic, lumbar and sacral areas. Person working with the DeepLabv3+ model produced segmentations with 0.925 Intersection over Union (IoU) and 0.920 Dice score along with 0.920 F1 score and an accuracy rate of 98.5%. The U-Net model achieved better performance when applied with a ResNet34 backbone by reaching an IoU of 0.953 alongside Dice score 0.951 F1 score 0.951 and accuracy reaching 98.6%. U-Net outperforms existing methods of spine segmentation by achieving superior results in area detection along with boundary definition accuracy. The research provides a thorough assessment of network designs and practical utility tests to support the development of automated spine segmentation applications that can work effectively in clinical settings.
Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior Shilpa Malge, Ram Kishun Lodhi International Journal of Mathematical Engineering and Management Sciences, 2025 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.
Quartic B-Spline Technique for Third-Order Linear Singularly Perturbed Boundary Value Problem with Discontinuous Source Term Shilpkala T. Mane, Ram Kishun Lodhi International Journal of Mathematical Engineering and Management Sciences, 2025 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.
Numerical Solution of Class of Third-Order Singularly Perturbed Delay Differential Equations via Quartic B-Spline Method Iaeng International Journal of Applied Mathematics, 2025
Cubic B-spline Technique for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term Iaeng International Journal of Computer Science, 2023
Numerical solution of second order self-adjoint singularly perturbed boundary value problems via quintic B-spline method with fitted mesh Mathematics in Engineering Science and Aerospace, 2023
Survey on numerical and analytical methods for singularly perturbed differential-difference equations Mathematics in Engineering Science and Aerospace, 2022
Quintic B-spline method for numerical solution of second-order singularly perturbed delay differential equations S Malge, RK Lodhi Computational Methods for Differential Equations 14 (1), 36-49 , 2026 2026
Numerical Solution of Newell-Whitehead-Segel Equation via Quintic B-spline Collocation Method NH Thube, RK Lodhi International Journal of Mathematical, Engineering and Management Sciences … , 2026 2026
Heat and Mass Transfer in MHD Maxwell Nanofluids over a Stretching Cylinder with Activation Energy and Chemical Reactions T Thumma, R Sandiri, SU Khan, RK Lodhi, K Ramesh Thermal Advances, 100126 , 2026 2026
Numerical Solution of Third-Order Singularly Perturbed Boundary Value Problems Via Quartic B-Spline Method HK Mishra, RK Lodhi Proceedings of the National Academy of Sciences, India Section A: Physical … , 2025 2025
Numerical solution of Hunter-Saxton type equation via exponential cubic B-spline collocation method NH Thube, RK Lodhi Boundary Value Problems 2025 (1), 1-14 , 2025 2025 Citations: 1
Leveraging DeepLabv3+ and U-Net Models for Accurate Spine Segmentation in CT Scans. R Singh, C Prabha, M Malik, RK Lodhi International Journal of Mathematical, Engineering & Management Sciences 10 (6) , 2025 2025
Numerical Solution of Class of Third-Order Singularly Perturbed Delay Differential Equations via Quartic B-Spline Method. S Malge, RK Lodhi IAENG International Journal of Applied Mathematics 55 (11) , 2025 2025
Numerical investigation of electroosmotic driven transport of MHD Casson fluid over an exponential stretching sheet using quartic B-spline method K Ramesh, KK Asogwa, RK Lodhi, MI Khan, M Medani, KA Gepreel, ... Numerical Heat Transfer, Part A: Applications 86 (19), 6901-6916 , 2025 2025 Citations: 13
Multiscale modeling of Jeffrey-Casson immiscible fluid dynamics under electrokinetic and magnetohydrodynamic effects M Bhaskar, KR Babu, SU Khan, RK Lodhi, F Mebarek-Oudina, M Sohail, ... Multiscale and Multidisciplinary Modeling, Experiments and Design 8 (9), 383 , 2025 2025 Citations: 6
Non-Polynomial Spline for Singularly Perturbed Differential-Difference Equation with Mixed Shifts and Layer Behavior S. Malge, R.K. Lodhi International Journal of Mathematical, Engineering and Management Sciences … , 2025 2025
Mathematical Modelling of Engineering Problems CK Chan, P Dumka, K Dodiya, DR Mishra, R Chauhan, C Sonawane, ... Journal homepage: http://iieta. org/journals/mmep 12 (6), 2195-2201 , 2025 2025
Quartic B-Spline Technique for Third-Order Linear Singularly Perturbed Boundary Value Problem with Discontinuous Source Term S T. Mane, RK Lodhi International Journal of Mathematical, Engineering and Management Sciences … , 2025 2025
SOLUTION OF TIME-FRACTIONAL BLACK–SCHOLES EQUATIONS VIA HOMOTOPY ANALYSIS SUMUDU TRANSFORM METHOD H Gao, RK Pandey, RK Lodhi, H Jafari Fractals 33 (06), 2540110 , 2025 2025 Citations: 2
Face Mask Recognition in a Street Camera Video Stream by Using Double CNN with Adaptive Particle Swarm Optimization R Bhandari, AK Jain, G Kaur, M Malik, D Nandan, OP Suthar, RK Lodhi Ingenierie des Systemes d'Information 30 (2), 449 , 2025 2025
Dynamics of sodium alginate-based ternary nanofluid flow over a stretching sheet with Al 2 O 3 , SiO 2 , and TiO 2 nanoparticles HD Akshatha, SM Sachhin, US Mahabaleshwar, RK Lodhi, K Ramesh Multiscale and Multidisciplinary Modeling, Experiments and Design 8 (1), 32 , 2025 2025 Citations: 20
A Hybrid PSO-AOQPIO Approach for Efficient Wind-Affected UAV Task Allocation and Path Optimization G Singh, R Walia, RK Lodhi 2024 IEEE 16th International Conference on Computational Intelligence and … , 2024 2024 Citations: 3
A novel quintic B-spline technique for numerical solution of fourth-order singularly-perturbed boundary value problems with discontinuous source terms ST Mane, RK Lodhi Boundary Value Problems 2024 (1), 167 , 2024 2024 Citations: 1
Quintic B-Spline Method for Numerical Solution of Second Order Singularly Perturbed Delay Differential Equations RK Lodhi, S Malge Computational Methods for Differential Equations , 2024 2024
Peristaltic propulsion of Jeffrey nanofluid with heat and electromagnetic effects: application to biomedicine K Ramesh, S Vemulawada, SU Khan, S Saleem, A Sharma, RK Lodhi, ... Multiscale and Multidisciplinary Modeling, Experiments and Design 7 (6 … , 2024 2024 Citations: 19
A Sixth-Order Cubic B-Spline Approach for Solving Linear Boundary Value Problems: An In-Depth Analysis and Comparative Study RK Lodhi, MS Darweesh, A Aydi, L Kolsi, A Sharma, K Ramesh Mathematics 12 (20), 3274 , 2024 2024 Citations: 1
MOST CITED SCHOLAR PUBLICATIONS
Computational analysis on radiative non-Newtonian Carreau nanofluid flow in a microchannel under the magnetic properties K Ramesh, F Mebarek-Oudina, AI Ismail, BR Jaiswal, AS Warke, ... Scientia Iranica 30 (2), 376-390 , 2023 2023 Citations: 97
Comparative study on electroosmosis modulated flow of MHD viscoelastic fluid in the presence of modified Darcy’s law RK Lodhi, K Ramesh Chinese Journal of Physics 68, 106-120 , 2020 2020 Citations: 48
Quintic B-spline method for solving second order linear and nonlinear singularly perturbed two-point boundary value problems RK Lodhi, HK Mishra Journal of Computational and Applied Mathematics 319, 170-187 , 2017 2017 Citations: 34
Solution of a class of fourth order singular singularly perturbed boundary value problems by quintic B-spline method RK Lodhi, HK Mishra Journal of the Nigerian Mathematical Society 35 (1), 257-265 , 2016 2016 Citations: 24
Dynamics of sodium alginate-based ternary nanofluid flow over a stretching sheet with Al 2 O 3 , SiO 2 , and TiO 2 nanoparticles HD Akshatha, SM Sachhin, US Mahabaleshwar, RK Lodhi, K Ramesh Multiscale and Multidisciplinary Modeling, Experiments and Design 8 (1), 32 , 2025 2025 Citations: 20
Peristaltic propulsion of Jeffrey nanofluid with heat and electromagnetic effects: application to biomedicine K Ramesh, S Vemulawada, SU Khan, S Saleem, A Sharma, RK Lodhi, ... Multiscale and Multidisciplinary Modeling, Experiments and Design 7 (6 … , 2024 2024 Citations: 19
Numerical solution of non-linear Bratu-type boundary value problems via quintic B-spline collocation method RK Lodhi, SF Aldosary, KS Nisar, A Alsaadi AIMS Mathematics 7 (4), 7257-7273 , 2022 2022 Citations: 16
Numerical investigation of electroosmotic driven transport of MHD Casson fluid over an exponential stretching sheet using quartic B-spline method K Ramesh, KK Asogwa, RK Lodhi, MI Khan, M Medani, KA Gepreel, ... Numerical Heat Transfer, Part A: Applications 86 (19), 6901-6916 , 2025 2025 Citations: 13
Septic B-spline method for second order self-adjoint singularly perturbed boundary-value problems RK Lodhi, HK Mishra Ain Shams Engineering Journal 9 (4), 2153-2161 , 2018 2018 Citations: 13
A NUMERICAL STUDY FOR NON-LINEAR MULTI-TERM FRACTIONAL ORDER DIFFERENTIAL EQUATIONS. SM Narsale, RK Lodhi, H Jafari Advanced Mathematical Models & Applications 9 (2) , 2024 2024 Citations: 9
Numerical analysis of electromagnetic squeezing flow through a parallel porous medium plate with impact of suction/injection P Jayavel, R Katta, RK Lodhi Waves in Random and Complex Media , 2022 2022 Citations: 9
Computational approach for fourth-order self-adjoint singularly perturbed boundary value problems via non-polynomial quintic spline RK Lodhi, HK Mishra Iranian Journal of Science and Technology, Transactions A: Science 42 (2 … , 2018 2018 Citations: 9
Two-parameter singular perturbation boundary value problems via quintic b-spline method HK Mishra, RK Lodhi Proceedings of the National Academy of Sciences, India Section A: Physical … , 2022 2022 Citations: 8
Cubic b-spline technique for numerical solution of singularly perturbed convection-diffusion equations with discontinuous source term S Mane, RK Lodhi IAENG International Journal of Computer Science 50 (2), 402-407 , 2023 2023 Citations: 7
Multiscale modeling of Jeffrey-Casson immiscible fluid dynamics under electrokinetic and magnetohydrodynamic effects M Bhaskar, KR Babu, SU Khan, RK Lodhi, F Mebarek-Oudina, M Sohail, ... Multiscale and Multidisciplinary Modeling, Experiments and Design 8 (9), 383 , 2025 2025 Citations: 6
Numerical Solution of Two-Parameter Singularly Perturbed Convection-Diffusion Boundary Value Problems via Fourth Order Compact Finite Difference Method RK Lodhi, BR Jaiswal, D Nandan, K Ramesh Mathematical Modelling of Engineering Problems 8 (5), 819-825 , 2021 2021 Citations: 4
A Hybrid PSO-AOQPIO Approach for Efficient Wind-Affected UAV Task Allocation and Path Optimization G Singh, R Walia, RK Lodhi 2024 IEEE 16th International Conference on Computational Intelligence and … , 2024 2024 Citations: 3
Single-Valued Neutrosophic CRITIC-Based ARAS Method for the Assessment of Sustainable Circular Supplier Selection AR Mishra, R Krishankumar, F Cavallaro, RK Lodhi, KS Ravichandran Decision Making Using AI in Energy and Sustainability: Methods and Models … , 2023 2023 Citations: 3
Quintic b-spline method for singularly perturbed fourth-order ordinary differential equations of reaction–diffusion type RK Lodhi, HK Mishra National Academy Science Letters 41 (6), 385-390 , 2018 2018 Citations: 3
Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems RK Lodhi Studia Universitatis Babes-Bolyai Matematica , 2018 2018 Citations: 3