Description:FIELD OF THE INVENTION

The present invention relates to the field of computational mathematics, numerical analysis, and thermal physics, with particular emphasis on the development of advanced numerical techniques for solving singular boundary-value problems (SBVPs) arising in the modeling of thermal explosion phenomena. Specifically, it pertains to a high-accuracy method and system employing high-order B-spline collocation to address the complexities inherent in nonlinear differential equations with singularities, which frequently occur in the simulation of chemically reactive systems, heat transfer processes, and combustion models. The invention integrates principles from functional analysis, spline theory, and nonlinear numerical methods to construct an efficient and robust computational framework for accurately approximating the solutions to SBVPs that are otherwise challenging to solve using conventional methods due to their steep gradients, boundary singularities, and sensitivity to initial approximations. This system is particularly applicable to problems governed by reaction-diffusion mechanisms, heat generation in exothermic reactions, and other thermally activated processes where precise temperature distribution prediction is crucial. By leveraging the local support and smoothness properties of high-order B-splines, along with strategically placed collocation points and adaptive mesh refinement, the invention offers enhanced resolution of singularities and improved convergence rates. It finds practical applications across various scientific and engineering domains, including chemical engineering, energy systems, safety analysis, and combustion theory. The invention bridges the gap between mathematical theory and real-world thermal modeling, thereby expanding the computational toolkit available for researchers and engineers dealing with singular thermal boundary-value problems.

Background of the proposed invention:

Singular boundary-value problems (SBVPs) frequently arise in the mathematical modeling of a variety of physical, biological, and chemical processes, especially those involving highly reactive systems such as thermal explosions. These problems are typically governed by nonlinear ordinary differential equations (ODEs) with boundary conditions imposed at singular points, often leading to complexities in analytical and numerical treatment. Thermal explosion theory, which models the sudden and self-accelerating increase in temperature due to exothermic chemical reactions within confined spaces, serves as a classical example of such SBVPs. The challenge in solving thermal explosion models stems from the inherent nonlinearity of the governing equations, the presence of singularities at boundaries (e.g., at the center of a spherical or cylindrical domain where the dependent variable and its derivatives may become unbounded), and the steep spatial gradients of the temperature and reaction rate profiles. Traditional numerical methods, such as finite difference, finite element, or shooting techniques, often struggle to maintain accuracy and stability near singularities or fail to converge altogether due to the sensitivity of the solution to initial guesses and discretization schemes. These issues are exacerbated in problems with strong nonlinearities, where small changes in input parameters can lead to vastly different outcomes, including thermal runaway or quenching. As thermal explosion phenomena are critical to many fields—including combustion science, chemical engineering, process safety, material synthesis, and energetic materials design—there is a strong demand for robust numerical methods that can accurately and efficiently handle such singular and nonlinear systems. Recent years have witnessed significant interest in spline-based methods, particularly B-spline and its higher-order extensions, due to their inherent smoothness, flexibility, and ability to represent complex functional behaviors with high accuracy. B-splines are piecewise polynomial functions defined over a partitioned domain, offering local support, smoothness, and the capacity to approximate functions and their derivatives to a high degree of precision. However, while lower-order B-splines (such as cubic or quartic) have been used in numerical applications, their effectiveness diminishes when addressing problems with steep gradients or boundary singularities. To address this gap, the proposed invention introduces a novel approach based on high-order B-spline collocation for solving SBVPs arising in thermal explosion problems. Collocation methods work by enforcing the differential equation at specific discrete points (collocation points) within the domain, transforming the problem into a system of algebraic equations that can be solved numerically. When integrated with high-order B-spline basis functions, the collocation method provides excellent approximation capabilities, particularly in regions of rapid solution variation or singular behavior. The strength of this approach lies in its ability to achieve spectral-like accuracy with fewer discretization points, thanks to the superior approximation properties of high-order B-splines. Moreover, this technique ensures that the solution and its derivatives are represented smoothly across the domain, which is essential for capturing the physical behavior of temperature and reaction rate distributions in thermal explosion models. To further improve the accuracy and reliability of the solution, the proposed invention incorporates an adaptive mesh refinement strategy that dynamically adjusts the grid spacing near singularities or regions with high gradient magnitudes. This feature allows the numerical method to concentrate computational effort where it is most needed, avoiding unnecessary computations in smooth regions and enhancing the overall efficiency of the solver. The invention also integrates a robust nonlinear solver—such as Newton-Raphson or quasi-linearization techniques—tailored to address the stiff and nonlinear character of the equations involved. This ensures stable and fast convergence of the numerical solution, even for highly reactive scenarios where traditional methods fail. The importance of this invention becomes evident when considering its wide applicability in real-world problems. In industrial safety engineering, for example, predicting the onset of thermal explosions in chemical storage tanks, reactors, or pipelines is essential for preventing catastrophic failures. Similarly, in combustion modeling, accurately resolving the ignition and flame propagation processes hinges on solving SBVPs that describe thermal runaway and energy release. In nuclear engineering, materials science, and catalysis, similar SBVPs appear when modeling thermal diffusion, radiation heat transfer, and reaction kinetics. Each of these applications demands a reliable, efficient, and high-fidelity numerical approach to ensure safety, optimize design, and enhance system performance. Additionally, the high-order B-spline collocation framework introduced in this invention is flexible and generalizable, allowing it to be extended to a wide class of singular problems beyond thermal explosions. It can be adapted to time-dependent problems, partial differential equations, and multidimensional domains, making it a powerful tool in the broader landscape of scientific computing. Furthermore, the modular nature of the method allows it to be implemented within existing computational software environments, facilitating its adoption by engineers, researchers, and practitioners. The proposed invention thus represents a significant advancement over existing techniques by combining the mathematical rigor of spline theory with practical computational strategies tailored for challenging boundary-value problems. It not only overcomes the limitations of conventional finite difference and shooting methods but also opens new avenues for the analysis and simulation of complex thermally reactive systems. By enabling accurate prediction of thermal behavior in systems with singularities and strong nonlinearities, this invention provides critical insights for decision-making, design optimization, and safety assurance in numerous technological domains. In essence, the invention fills a crucial void in the computational modeling of thermal explosion phenomena by offering a high-order B-spline collocation system that is mathematically sound, numerically stable, computationally efficient, and broadly applicable. It builds upon the foundational concepts of numerical approximation and differential equation theory while pushing the boundaries of their application to address some of the most demanding problems in modern thermal analysis and reaction engineering.

Summary of the proposed invention:

The proposed invention presents a comprehensive and innovative method and system for solving singular boundary-value problems (SBVPs) that frequently arise in thermal explosion models through the implementation of a high-order B-spline collocation framework. Thermal explosions are a class of nonlinear physical phenomena characterized by self-accelerating heat release due to exothermic chemical reactions in confined media, where the temperature rises rapidly leading to potential hazards such as combustion, material degradation, or system failure. The governing equations of such systems are typically second-order nonlinear ordinary differential equations with singularities at the boundaries, making them notoriously difficult to solve accurately using conventional numerical methods such as finite difference, shooting methods, or basic spline approaches. These conventional techniques often suffer from low convergence rates, numerical instability, and an inability to capture the sharp gradients or singular behaviors near the boundary, especially in highly stiff problems. In contrast, the proposed invention addresses these challenges by developing a robust high-order collocation scheme that employs quintic or higher-order B-spline basis functions to approximate the solution over a discretized domain. B-splines, known for their smoothness, local support, and high accuracy, are utilized here not merely for interpolation but as functional bases within the collocation framework, enabling the method to achieve enhanced precision in representing the underlying physical behavior of the system. The method discretizes the domain using a non-uniform mesh that adapts to regions of steep gradient or singularity, allowing for better resolution without unnecessarily increasing the computational burden across the entire domain. The collocation points are strategically chosen to coincide with areas of high solution sensitivity, often near the singular boundaries, thus ensuring that the numerical system formed is both sparse and highly accurate. The core numerical engine incorporates nonlinear solvers such as the Newton-Raphson method or quasi-linearization techniques that are embedded into the B-spline collocation scheme to effectively tackle the nonlinearities inherent in thermal explosion problems. This combination ensures rapid convergence, even in scenarios involving extreme reaction rates and tight thermal coupling. Furthermore, the system supports an automated adaptive refinement strategy where error estimation is used to guide the dynamic redistribution of collocation points, further improving the convergence and stability of the method in the presence of singularities or stiff gradients. The entire system is implemented in a modular computational framework that allows for customization of the B-spline order, domain geometry, boundary conditions, and reaction kinetics, making it applicable to a wide variety of thermal explosion models including planar, cylindrical, and spherical geometries. Importantly, the invention is not limited to stationary problems but can be extended to transient or time-dependent thermal explosion scenarios by coupling with time-integration methods, thereby broadening its scope of applicability. This system not only addresses the limitations of existing numerical techniques but also significantly enhances the fidelity of modeling real-world thermal explosion phenomena in chemical reactors, energetic materials, catalytic processes, and other reactive thermal systems. Comparative performance analyses against traditional finite difference methods, shooting methods, and lower-order spline techniques demonstrate that the proposed invention achieves superior accuracy with fewer computational nodes and reduced solution time, especially for problems involving boundary singularities or highly nonlinear source terms. Moreover, the B-spline-based collocation system provides inherent advantages in terms of smoothness of the solution and its derivatives, which is critical for post-processing tasks such as stability analysis, bifurcation detection, and sensitivity studies. The invention is particularly valuable for applications in chemical engineering, combustion modeling, safety analysis, and thermal system design, where accurate and efficient prediction of temperature and reaction profiles is vital. For example, in chemical process industries, the ability to model and predict thermal runaway conditions using this system can help in designing safer reactors and storage facilities. In energy and propulsion systems, the method can contribute to the accurate simulation of ignition and combustion in confined chambers, aiding in performance optimization. Additionally, in the context of academic research, the method serves as a powerful computational tool for investigating theoretical aspects of thermal explosion and reaction-diffusion systems. The invention also includes provisions for integration with modern computational environments and software libraries, allowing users to incorporate the B-spline collocation engine into broader simulation workflows or couple it with multi-physics solvers for holistic modeling. Furthermore, due to its high accuracy and adaptability, the system can be utilized in inverse problems, where thermal explosion parameters need to be inferred from experimental data, thus supporting diagnostic and monitoring applications. The high-order B-spline collocation approach introduced by this invention marks a significant improvement in numerical treatment of SBVPs, combining the mathematical rigor of spline theory with practical advancements in adaptive numerical strategies. It transforms a complex, previously difficult class of thermal problems into tractable computational tasks, thereby contributing to safer designs, improved thermal management, and better theoretical understanding. In essence, this invention delivers a versatile, scalable, and highly accurate solution to a longstanding computational challenge, bridging the gap between advanced numerical mathematics and real-world thermal engineering problems. It empowers engineers, scientists, and researchers with a powerful new tool for analyzing and solving nonlinear thermal systems characterized by singular boundary conditions, offering significant benefits in accuracy, efficiency, flexibility, and applicability across a wide range of scientific and industrial domains.

Brief description of the proposed invention:

The proposed invention introduces a high-precision numerical method and computational system specifically designed to solve singular boundary-value problems (SBVPs) that commonly arise in the mathematical modeling of thermal explosion phenomena. These types of problems typically involve second-order nonlinear ordinary differential equations (ODEs) defined over a finite domain with boundary conditions that are singular in nature, meaning the solution or its derivatives may become undefined or unbounded at one or both endpoints of the domain. Such conditions are characteristic of thermal explosion models, where heat generated from exothermic chemical reactions within confined reactive systems accumulates faster than it can dissipate, leading to a rapid rise in temperature known as thermal runaway. Traditional numerical methods like finite difference methods, shooting methods, or low-order spline techniques often fall short when applied to these problems, suffering from convergence difficulties, loss of accuracy near singularities, and poor stability in stiff regimes. In contrast, the proposed invention leverages the power of high-order B-spline collocation to provide a more accurate, stable, and computationally efficient solution. B-splines are piecewise polynomial functions with desirable numerical properties, such as compact support, smoothness, and the ability to approximate complex functions with high fidelity. In this invention, high-order B-splines (such as quintic, septic, or higher) are used as basis functions to approximate the unknown solution of the SBVP across a discretized domain. The domain is divided into non-uniform intervals, with finer resolution near the boundaries or regions of steep gradients, to capture the singular behavior more precisely. A set of collocation points is strategically selected within the domain, including regions close to the singularities, and the governing differential equation is enforced to hold true at these points. This leads to a system of nonlinear algebraic equations, which are solved using iterative numerical solvers like Newton-Raphson or quasi-linearization techniques, embedded within the B-spline framework. This ensures that the nonlinear nature of the problem is tackled effectively while maintaining numerical stability and rapid convergence. The high-order B-spline basis functions also provide smooth approximations not only to the solution itself but to its derivatives, which is important in thermal analysis where heat flux and reaction rates depend on gradients. One of the unique features of this invention is the integration of an adaptive mesh refinement (AMR) module that assesses local approximation errors and dynamically adjusts the distribution of collocation points and knot locations in the B-spline basis. This adaptivity allows the system to concentrate computational effort in regions where the solution exhibits sharp transitions or singular behavior, without expending unnecessary resources in smoother regions, thus improving both efficiency and accuracy. The method is designed to be modular and extensible, allowing for easy implementation of different types of boundary conditions (Dirichlet, Neumann, Robin), geometries (planar, cylindrical, spherical), and nonlinear source terms specific to various thermal explosion scenarios. For instance, models based on Frank-Kamenetskii theory or Semenov’s criteria for thermal ignition can be easily accommodated. Additionally, the system can be configured for steady-state or transient problems by incorporating time-discretization schemes such as implicit Euler or Crank-Nicolson for temporal evolution. The computational system developed as part of the invention includes a user-friendly interface for specifying the problem parameters, B-spline order, collocation strategy, and convergence criteria, making it accessible to both expert users in numerical analysis and application-focused engineers or scientists. It also features built-in visualization tools to plot the solution, its first and second derivatives, and error estimates, thereby facilitating interpretation and analysis. The invention’s architecture supports integration with other numerical packages or simulation environments, allowing it to be used as a component in larger multi-physics simulations, for example, those involving fluid flow, mass transfer, or structural deformation coupled with thermal explosion behavior. Beyond solving the direct SBVP, the system is capable of performing sensitivity analysis, bifurcation tracking, and stability assessments, which are critical in understanding the onset of explosive behavior under varying conditions. Comparative benchmarks conducted as part of the invention’s development show that the high-order B-spline collocation method significantly outperforms traditional methods in terms of accuracy per computational cost, especially for problems with strong nonlinearities and boundary singularities. The results demonstrate that fewer collocation points are required to achieve the same or better accuracy than lower-order schemes, and the adaptivity mechanism further reduces computational overhead by intelligently allocating resources. The invention finds broad application in fields such as chemical process engineering, combustion science, safety engineering, and environmental modeling. In industrial applications, it can be used to model heat buildup in chemical reactors, predict ignition thresholds in fuel storage systems, or assess the thermal stability of novel energetic materials. In academic and research settings, the invention provides a valuable tool for exploring theoretical aspects of thermal explosion and nonlinear dynamics, offering insights into solution multiplicity, critical parameters, and the effects of boundary conditions or spatial dimensions on the explosion threshold. Furthermore, the system is highly suitable for educational purposes, as it offers a transparent and customizable platform for students and researchers to experiment with complex thermal models and understand the behavior of SBVPs through interactive simulation. The invention may also be extended or adapted to solve other classes of singular or stiff problems in science and engineering, such as reaction-diffusion systems, biological pattern formation, electrochemical modeling, or astrophysical phenomena, highlighting its versatility and robustness. Its efficient numerical engine, combined with a mathematically rigorous yet practically implementable framework, enables users to tackle problems that were previously inaccessible or required excessive computational effort. In conclusion, this invention provides a transformative advancement in the numerical solution of singular boundary-value problems in thermal explosion modeling, combining the theoretical strengths of high-order B-splines with practical innovations in collocation, adaptivity, and nonlinear solution techniques. It bridges the longstanding gap between mathematical sophistication and engineering applicability, delivering a reliable, accurate, and efficient solution strategy that meets the growing demands of modern thermal analysis and reactive system design. By offering a general-purpose, high-fidelity computational platform, the proposed system redefines the standards for solving SBVPs in both academic and
industrial contexts.
, Claims:We Claim:

1. A method for numerically solving singular boundary-value problems (SBVPs) arising in thermal explosion models, comprising the use of high-order B-spline basis functions within a collocation framework to approximate the solution of nonlinear ordinary differential equations with singular boundary conditions.

2. The method of claim 1, wherein the B-spline functions are of quintic order or higher, enabling smooth and accurate approximation of both the solution and its derivatives over a discretized domain with singularities.

3. The method of claim 1, further comprising the step of selecting non-uniform collocation points that are strategically placed near singularities or regions with steep solution gradients to improve local accuracy and convergence.

4. The method of claim 1, wherein an adaptive mesh refinement (AMR) algorithm dynamically adjusts the discretization of the domain based on real-time error estimates to concentrate computational effort near singular or high-gradient regions.

5. The method of claim 1, wherein the resulting system of nonlinear algebraic equations is solved using an iterative solver selected from a group consisting of Newton-Raphson, quasi-linearization, and other nonlinear solvers tailored for stiff problems.

6. A system for solving thermal explosion SBVPs, comprising a computational module for defining boundary conditions, selecting B-spline order, applying collocation constraints, solving nonlinear algebraic systems, and visualizing solution profiles.

7. The system of claim 6, wherein the computational module allows users to input problem-specific data, including thermal explosion parameters, boundary types (Dirichlet, Neumann, Robin), and domain geometries (planar, cylindrical, or spherical).

8. The system of claim 6, further comprising an error estimation and refinement module configured to identify areas of high approximation error and adjust the knot vector or collocation points to enhance solution precision.

9. The method of claim 1, wherein the collocation framework is extendable to time-dependent thermal explosion problems through integration with time-marching schemes such as implicit Euler or Crank-Nicolson methods.

10. The method and system as claimed in any of the preceding claims, wherein the solution output includes not only the temperature profile but also derivative fields (e.g., heat flux and reaction rate) and diagnostic plots for physical interpretation and design applications.