Numerical simulation of soliton solutions of nonlinear Fitzhugh-Nagumo equation by using LOOCV with exponential B-spline with Significant Applications in Neurosciences

doi:10.21203/rs.3.rs-5323419/v1

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Numerical simulation of soliton solutions of nonlinear Fitzhugh-Nagumo equation by using LOOCV with exponential B-spline with Significant Applications in Neurosciences

https://doi.org/10.21203/rs.3.rs-5323419/v1

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This study focuses on solving the one-dimensional nonlinear Fitzhugh-Nagumo (FHN) equation using a novel technique called the “Exponential modified cubic B-spline differential quadrature method” combined with “leave-one-out cross-validation”. The inclusion of leave-one-out cross-validation (LOOCV) is essential for finding the optimal value of the parameter $\:\lambda\:$, which is a key component in the exponential modified cubic B-spline basis functions, thereby enhancing the accuracy and robustness of the results. By incorporating this unique combination of LOOCV and the exponential modified cubic B-spline differential quadrature method, the research introduces a new computational approach that could be of considerable interest to scholars in the field. This method has been applied to four different examples of the Fitzhugh-Nagumo equation, with outcomes detailed in tables and figures. This paper presents the methodology and results of a study on the equation, emphasizing its significance and applications in neuroscience. The Fitzhugh-Nagumo model is highlighted as a versatile tool across various scientific, engineering, and mathematical fields, with a particular focus on its role in understanding the complex dynamics of neural systems and its potential impact on future research and real-world problems.

Differential quadrature method

leave-one-out cross-validation

exponential cubic B-spline

Fitzhugh-Nagumo equation

soliton solutions

Neurosciences

In 1952, Hodgkin and Huxley [1] introduced a classical single-neuron model, successfully simulating a wide range of neuronal behaviors. However, its complexity, involving numerous dynamic variables, presented challenges for network simulations. To address this, Fitzhugh [2] simplified the model in 1961, proposing the Fitzhugh-Nagumo (FHN) neuron, which Nagumo et al. [3] further refined in 1962. The FHN model, a streamlined version of the Hodgkin-Huxley model, focuses on fundamental aspects of excitable systems and provides insights into threshold behavior, recovery processes, and oscillations. The Fitzhugh Nagumo (FNE) equation is a type of non-linear partial differential equation presented as:

$$\:{u}_{t}={u}_{xx}+u\left(u-\zeta\:\right)\left(1-u\right),\:\:\:\:0<\zeta\:<1$$

1.1

Here $\:u(\text{x},\text{t})\:$represents the transmission of electrical potential through the cell membrane in application as a model to the propagation of nerve impulses. The FHN Eq. (1.1) with the following initial and boundary conditions:

$$\:u\left(x,0\right)=f\left(x\right),\:\:x\in\:\left[a,b\right],\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1.2\right)$$

$$\:u\left(a,t\right)=\:{g}_{1}\left(t\right),\:\:\:u\left(b,t\right)=\:{g}_{2}\left(t\right),\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1.3\right)\:$$

Solitons, stable localized wave packets that maintain their shape while traveling at a constant speed, arise in nonlinear systems and represent solutions to certain partial differential equations [4]. In the context of the FHN model, soliton-like solutions can represent the propagation of action potentials along axons, crucial for understanding how signals travel long distances in neural and other excitable media [5]. Triki and Wazwaz [6] considered a generalized FHN equation with time-varying coefficients and a linear dispersion term. Using a solitary wave ansatz and the tanh method, the authors have derived new soliton solutions, with physical parameters as functions of the time-dependent coefficients. They also presented conditions for the existence and uniqueness of solitons.

The FHN model is a cornerstone in the study of excitable systems, providing a simplified yet powerful framework for understanding the dynamics of neurons and other biological and chemical systems. Its applications in neuroscience, cardiac electrophysiology, and chemical oscillations highlight its versatility and importance. The FHN model is particularly valuable for studying wave-like propagation of action potentials, essential for understanding nerve impulse transmission [7] and cardiac dynamics [8]. The model’s nonlinear dynamics allow it to exhibit complex phenomena such as bifurcations and chaos under specific conditions [9]. Moreover, the FHN model’s adaptability extends to modeling excitable media and reaction-diffusion systems, providing valuable insights into complex spatial patterns and self-organized structures. This capability supports its significant contributions to mathematical biology, facilitating the modeling of population dynamics and disease spread, thereby aiding in epidemiological studies. Additionally, its capacity to simulate bifurcations, chaos, and other nonlinear dynamics makes it indispensable for investigating various dynamical systems, applying control theory, and addressing engineering tasks such as signal processing and electronic circuit simulation.

1.1 Applications of the Fitzhugh-Nagumo Model in Neurosciences

This model provides a simplified yet powerful framework for understanding various phenomena in the neural sciences. In neuroscience, the FHN model plays a pivotal role in modeling action potentials, enabling researchers to explore neural responses to stimuli and signal propagation through neural networks. It is crucial for simulating and analyzing neural oscillations, shedding light on rhythmic firing and pattern generation within the brain. The model also extends its applications to cardiac electrophysiology by simulating action potentials in cardiac cells and studying wave propagation in cardiac tissues, critical for understanding arrhythmias. The applications of the FHN Model are presented in Fig. 1.1.

1.1.1 Understanding Neuronal Excitability

The FHN model serves as an essential tool for studying the excitability of neurons. By reducing the complexity of the Hodgkin-Huxley model, the FHN equations allow researchers to explore how neurons respond to stimuli and how action potentials are generated and propagated. This understanding is crucial for unraveling the basic mechanisms underlying neural signaling and communication. Bisquert [10] presents an innovative frequency domain analysis of the FHN model, which simplifies neuron excitability dynamics compared to the Hodgkin-Huxley model. By translating the FHN model’s differential equations into an equivalent circuit model, the study utilizes impedance spectroscopy to identify Hopf bifurcations and spiking regimes. This approach highlights the essential components of a neuron-capacitor, inductor, and negative differential resistance-and suggests potential applications in developing memristor-based artificial neural networks.

1.1.2. Analysis of Action Potential Dynamics

One of the primary applications of the FHN model is the analysis of action potential dynamics. The model effectively captures the generation of action potentials, their propagation, and the refractory periods following a spike. Researchers use the FHN model to simulate and study the conditions under which neurons fire, the effects of different parameters on firing patterns, and how neurons transition between resting and active states. Naderi, Yazdanpanah, Azemi, and Roaia [11] present an adaptive feedback linearization controller for tracking normal action potentials using the FHN model. The controller stabilizes unstable rhythms and tracks normal action potentials using output feedback, with an adaptive observer estimating unknown parameters and states. Simulation results confirm the method’s effectiveness, highlighting its potential in controlling cardiac arrhythmia.

1.1.3. Wave Propagation in Neural Tissues

The FHN model is instrumental in examining wave propagation in neural tissues. It provides insights into how electrical signals travel through axons and across neural networks. This application is particularly relevant in understanding phenomena such as signal transmission in the brain, the spread of epileptic seizures, and the mechanisms of wave-like activities observed in cortical tissues. Verisokin et al. [7] extend the FHN system to model volume transmission in neural tissues, focusing on wave propagation and noise-sustained patterns. Their study reveals various spatio-temporal behaviors, including self-organizing bursts and slow-moving wave segments, highlighting the importance of volume transmission in neural communication.

1.1.4. Pattern Formation and Oscillations

In neural networks, the FHN model helps to study pattern formation and oscillatory behavior. It is used to investigate how coherent patterns of activity, such as waves or oscillations, emerge in neural populations. This is important for understanding various brain functions, including rhythmic activities associated with sleep, motor control, and cognitive processes. Zheng and Shen [9] investigated pattern formation in the FHN model, focusing on the effect of diffusion. They conducted linear stability analysis of local equilibrium to determine conditions for Turing bifurcation, Hopf bifurcation, and oscillatory instability boundaries. Taha, Tabib, and Kofane [10] investigate Hopf and Pitchfork bifurcations in the FHN neuron model with diffusion relaxation, focusing on pattern formation. They analytically establish these bifurcations, reduce the model to the complex Ginzburg-Landau equation, and conduct numerical simulations to compare with analytical predictions.

1.1.5. Modeling Cardiac Dynamics

Beyond the nervous system, the principles of the FHN model are applied to cardiac dynamics. The model’s equations are analogous to those describing excitable cardiac cells, making it useful for studying heart rhythms and arrhythmias. This interdisciplinary application highlights the FHN model’s broader relevance to excitable media in biological systems. Shuaiby, Hassan, and El-Melegy [8] present a study on modeling and simulating the electrical activity of human cardiac tissues using the finite element method. They employ a monodomain model coupled with a modified FHN model to simulate cardiac excitation and isotropic propagation. The research highlights the challenges of numerically solving the nonlinear and rapidly changing ordinary differential equation (ODE) systems that describe the electrical behavior of cardiac cell membranes.

1.1.6. Exploring Neurophysiological Disorders

The FHN model is also employed to explore neurophysiological disorders. For example, it helps in understanding the mechanisms of epilepsy, where abnormal, excessive neural activity occurs. By simulating the conditions leading to seizure-like activities, researchers can investigate potential therapeutic interventions and predict the effects of various treatments [7], [12].

1.1.7. Synaptic Interactions and Network Dynamics

The FHN model extends to the study of synaptic interactions and network dynamics. By incorporating synaptic coupling into the model, researchers can simulate how neurons interact within a network, how these interactions affect overall network behavior, and how synchronization and desynchronization phenomena occur. Breakspear and Stam [12] investigates the multiscale dynamics of neural networks, demonstrating how synaptic interactions influence overall network behavior. It highlights the use of models like the FHN model to study synchronization and desynchronization phenomena in neural systems.

Due to its critical applications, the FHN equation has been extensively solved both numerically and analytically by numerous researchers. Uyulan [13] developed a polynomial differential quadrature-based numerical scheme to simulate nerve pulse propagation in the spatial FHN model. This research, situated within the field of mathematical neuroscience, addresses the nonlinear dynamics connecting neurons to facilitate complex information production and transport in the brain. Mittal et al. [14] presented an improved Chebyshev-Gauss-Lobatto pseudo-spectral approximation for nonlinear Burger-Huxley and Fitzhugh-Nagumo equations. Their study employed a spectral method in both time and space, utilizing Chebyshev Gauss-Lobatto points to achieve spectral accuracy. Mehta et al. [15] developed a novel three-step second derivative block method coupled with fourth-order standard compact finite difference schemes for solving time-dependent nonlinear partial differential equations. They applied this method to well-known problems such as the FHN equation and the Burger’s equation to demonstrate its effectiveness. Jiwari et al. [16] applied the polynomial differential quadrature method to numerically solve the generalized FHN equation with time-dependent coefficients in one-dimensional space. Abbasbandy [5] applied the homotopy analysis method (HAM) to derive soliton solutions for the FHN equation. HAM, known for its effectiveness in obtaining exact and series solutions, was employed to analyze and provide insights into the dynamics of the equation. Ali et al. [17] developed a novel Galerkin finite element method approach to numerically compute solutions for the FHN equation. Ramos et al. [18] introduced an optimized hybrid block method combined with a modified cubic B-spline method for solving nonlinear FHN equation. Singh [19] presents a mixed-type discontinuous Galerkin approach to solve the generalized FHN reaction-diffusion model. Zheng and Shen [9] employed weak nonlinear multiple scales analysis and Taylor series expansion methods for solving nonlinear FHN equation.

Numerical methods play a crucial role in solving differential equations, providing computational solutions for complex mathematical problems. Researchers have access to a variety of mathematical tools to find numerical solutions to differential equations. One popular numerical technique is the differential quadrature method (DQM), which has been used successfully to solve a range of partial differential equations (PDEs). DQM, which was first presented by Bellman and Casti in the 1970s [20], [21], DQM underwent further development in the 1980s [22], establishing itself as a valuable tool for addressing differential equations in physical and engineering sciences [23]. This method has proven effective in solving differential equations in diverse fields, including biosciences, fluid dynamics, static and dynamic structural mechanics, transport processes, and lubrication mechanics. Professor Chang Shu [24] wrote a book on DQM, documenting its use in general engineering up to 1999. Quan and Chang [25] later enhanced DQM by refining the weighting coefficient formulation, a key aspect of the method. This advancement allowed for improved estimation of weighting coefficients using various test functions, such as spline functions, Lagrange interpolation polynomials, and sinc functions. DQM has since been applied to solve a range of PDEs using different basis functions, including quartic B-spline functions, modified cubic B-spline functions, and exponential cubic B-spline functions [26]–[30]. This versatility has made DQM a well-regarded method for addressing complex differential equations in numerous scientific and engineering contexts.

Exponential modified cubic B-spline (Expo-MCB) basis functions are frequently used in computational mathematics to solve differential equations due to their flexibility and accuracy. It has been successfully applied to a variety of equations, including the Burger’s equation [31], multi-dimensional convection-diffusion equations [32], the Sine-Gordon equation [33], [34], Fisher’s reaction-diffusion equation [35], the telegraph equation [36], and the nonlinear Schrodinger equation [37]. Despite their wide application, the differential quadrature method (DQM) with exponential modified cubic B-spline (Expo-MCB) basis functions (Expo-MCB-DQM) remains relatively rare in the literature. This lack of prevalence can be attributed to the inherent challenge of selecting the optimal parameter in Expo-MCB basis functions. Many researchers traditionally assigned this parameter arbitrary values, resulting in unstable or inconsistent outcomes. The randomness in choosing the parameter made it difficult to ensure reliable solutions, limiting the broader adoption of Expo-MCB-DQM.

To overcome this challenge, our research introduces a novel approach that uses leave-one-out cross-validation (LOOCV) to determine the optimal value for this parameter. By applying LOOCV, we aim to increase the stability and accuracy of solutions derived from Expo-MCB-DQM. This technique involves systematically removing one data point at a time from the dataset and testing the impact of different parameter values, allowing us to identify the best parameter setting for reliable results. Previous studies that used Expo-MCB basis functions often neglected this crucial step, leading to variability in the solutions. By contrast, our research seeks to establish a more robust methodology for solving differential equations with Expo-MCB-DQM. Through LOOCV, we provide a structured approach to parameter optimization, ensuring that the resulting solutions are not only accurate but also reproducible. This advancement has the potential to make Expo-MCB-DQM a more widely accepted technique in computational mathematics and related fields.

1.1 Leave-One-Out Cross-Validation (LOOCV)

Leave-one-out cross-validation (LOOCV) is a resampling technique frequently used in statistics and machine learning to enhance the robustness and precision of statistical models. The process involves creating multiple training and validation sets from a single dataset. In LOOCV, each data point is systematically set aside as a validation set, while the rest of the data is used to train the model. This step is repeated such that every data point serves as a validation instance exactly once, resulting in as many iterations as there are observations in the dataset. LOOCV offers an unbiased assessment of a model’s performance since every observation is used for both training and validation. It is particularly beneficial with smaller datasets because it maximizes data utilization, ensuring that no data point is wasted. Additionally, LOOCV can be especially useful when datasets have imbalanced classes or specific structural dependencies among observations.

Rippa [38] proposed using LOOCV to find the optimal value for the shape parameter in mathematical interpolation. Arora and Bhatia [39] combined the radial basis function (RBF) pseudo-spectral method with leave-one-out cross-validation (LOOCV) to address geometric flexibility limitations, showing its effectiveness for Fisher’s equation. This method uses LOOCV to determine the optimal shape parameter, which is crucial for achieving accurate results. This hybrid technique provides high accuracy and wider applicability for solving partial differential equations. LOOCV is a robust technique for assessing a model's performance because it uses every observation for both training and testing, reducing bias in the evaluation process. This technique has been effectively employed in the study to select the optimal shape parameter in radial basis functions (RBF) [40].

LOOCV serves as a valuable technique in a range of applications, offering a rigorous method for model validation and parameter tuning, especially when dealing with complex data structures or smaller datasets where every observation counts.

This paper presents an innovative modification to existing numerical techniques. Specifically, it develops a hybrid scheme to achieve more accurate numerical solutions for the FHN equation. This approach employs the exponential modified cubic B-spline basis function with the differential quadrature method, augmented by the optimization technique known as leave-one-out cross-validation (LOOCV). By integrating these methods, we establish a robust and accurate framework for solving the FHN equation, which holds significant importance across scientific and engineering disciplines, particularly in neuroscience.

The paper is organized as follows: Section 2 describes the numerical method used to implement leave-one-out cross-validation (LOOCV) in the context of exponential cubic B-spline with the differential quadrature method. Section 3 tests the method’s accuracy and reliability by applying it to numerical problems derived from the Fitzhugh-Nagumo equation. Finally, Section 4 discusses the results and provides insights into the findings.

2.1 The Differential Quadrature Method with B-spline Basis Functions Algorithm

The differential quadrature method (DQM) is a numerical approach for solving nonlinear partial differential equations (PDEs), utilizing B-spline functions as the basis functions. The algorithm for implementing DQM with B-spline basis functions can be described as follows:

Step 1: Discretize the Domain Start by dividing the domain of the nonlinear PDE into a series of discrete points. Ideally, these points should be evenly spaced to facilitate easier computation.

Step 2: Select Basis Functions Choose suitable basis functions to represent the unknown function and its derivatives at the discrete points. This step is crucial for accurately capturing the underlying structure of the PDE.

Step 3: Approximate the Unknown Function Use a weighted sum of the function values at the discrete points to approximate the unknown function and its derivatives. In this method, the weighting coefficients are derived from exponential cubic B-spline basis functions.

Step 4: Derivative Calculation With the approximation from Step 3, calculate the derivatives of the unknown function at each discrete point.

Step 5: Construct a System of Equations Substitute the derived derivatives into the original PDE, transforming it into a system of ordinary differential equations (ODEs). This transformation facilitates easier solution of the nonlinear PDE.

Step 6: Solve the System of Equations Finally, solve the resulting system of algebraic equations to find the values of the unknown function at the discrete points.

This algorithm streamlines the process of solving complex nonlinear PDEs using DQM with B-spline basis functions. By following these steps, you can accurately approximate and evaluate the derivatives and ultimately solve the system of equations to obtain reliable solutions.

2.2 Exponential modified cubic B-spline differential quadrature method

DQM is a numerical method for approximating the function derivative as a linear sum of the function value at the discrete node points inside the problem’s solution space. This technique takes into consideration the grid distribution, where a given interval $\:\left[a,b\right]$ is divided into a set of discrete grid points $\:a={x}_{1}<{x}_{2}<\dots\:<{x}_{N}=b$.

If the function $\:u\left(x\right)$ is sufficiently smooth within the region of the solution, the value of the derivatives at the discrete points $\:{x}_{i}\:$can be written in the form as follows:

$$\:\frac{{d}^{\left(r\right)}u}{d{x}^{\left(r\right)}}{\left.\right|}_{{x}_{i}}=\:{\sum\:}_{j=1}^{N}{p}_{i,j}^{\left(r\right)}u\left({x}_{j}\right),\:\:\:\:\:i=\text{1,2},\dots\:,N,\:\:\:\:r=\text{1,2},\dots\:,N-1$$

2.1

here $\:r$ signifies the order of derivative, $\:{p}_{i,j}^{\left(r\right)}$ are the weighing coefficients and $\:N$ are the total points consider for approximation solution in the domain.

DQM aims to estimate weighting coefficients by using a set of basis functions that span the domain. To calculate these weighting coefficients, different types of basis functions can be used depending on the specific requirements of the problem.

This research employs the exponential form of B-spline in the third degree as the basis functions for calculating the weighting coefficients. The exponential cubic B-spline basis functions are defined as follows [31]:

$$\:{C}_{m}\left(x\right)=\frac{1}{{h}^{3}}\left\{\begin{array}{cc}{\alpha\:}_{3}\:\left({x}_{m-2}-x\right)-\frac{{\alpha\:}_{3}}{\lambda\:\left(\text{sinh}\left(\lambda\:\left({x}_{m-2}-x\right)\right)\right)},&\:x\in\:\left[{x}_{m-2},{x}_{m-1}\right)\\\:{\alpha\:}_{1}+{\alpha\:}_{2}\left({x}_{m}-x\right)+{\alpha\:}_{4}{e}^{\lambda\:\left({x}_{m}-x\right)}+{\delta\:}_{1}{e}^{-\lambda\:\left({x}_{m}-x\right)},&\:x\in\:\left[{x}_{m-1},{x}_{m}\right)\\\:{\alpha\:}_{1}+{\alpha\:}_{2}\left(x-{x}_{m}\right)+{\alpha\:}_{4}{e}^{\lambda\:\left(x-{x}_{m}\right)}+{\delta\:}_{1}{e}^{-\lambda\:\left(x-{x}_{m}\right)},&\:x\in\:\left[{x}_{m},{x}_{m+1}\right)\\\:{\alpha\:}_{3}\left(x-{x}_{m+2}\right)-\frac{{\alpha\:}_{3}}{\lambda\:\left(\text{sinh}\left(\lambda\:\left(x-{x}_{m+2}\right)\right)\right)},&\:x\in\:\left[{x}_{m+1},{x}_{m+2}\right)\end{array}\right.$$

2.2

$$\:\text{w}\text{h}\text{e}\text{r}\text{e}\:h={x}_{n}-{x}_{n-1}\:\text{f}\text{o}\text{r}\:\text{a}\text{l}\text{l}\:n.$$

$$\:{{\alpha\:}}_{1}=\frac{\lambda\:h{c}_{1}}{\lambda\:h{c}_{1}-{c}_{2}}\:,\:{{\alpha\:}}_{2}=\frac{\lambda\:}{2}\left(\begin{array}{c}\frac{{c}_{1}\left({c}_{1}-1\right)+{{c}_{2}}^{2}}{\left(\lambda\:h{c}_{1}-{c}_{2}\right)\left(1-{c}_{1}\right)}\end{array}\right),\:{{\alpha\:}}_{3}=\frac{\lambda\:}{2\left(\lambda\:h{c}_{1}-{c}_{2}\right)}\:,\:{{\alpha\:}}_{4}\:=\frac{1}{4}\left(\begin{array}{c}\frac{\left(1-{c}_{1}+{c}_{2}\right){e}^{-\lambda\:h}-{c}_{2}}{\left(\lambda\:h{c}_{1}-{c}_{2}\right)\left(1-{c}_{1}\right)}\end{array}\right),$$

$$\:{\delta\:}_{1}=\:\frac{1}{4}\left(\begin{array}{c}\frac{\left(-1+{c}_{1}+{c}_{2}\right){e}^{\lambda\:h}-{c}_{2}}{\left(\lambda\:h{c}_{1}-{c}_{2}\right)\left(1-{c}_{1}\right)}\end{array}\right),\:{c}_{1}=\text{cosh}\left(\lambda\:h\right),\:{c}_{2}=\text{sinh}\left(\lambda\:h\right).$$

The parameter $\:(\lambda\:$) need to be obtained for the minimum error, which is necessary for finding the solutions.

The numerical values of the exponential cubic B-spline functions $\:{C}_{m}\left(x\right)$ and their derivatives at different nodal points can be obtained as follows:

$$\:{C}_{m}\left({x}_{m-1}\right)={C}_{m}\left({x}_{m+1}\right)=\frac{{c}_{2}-\lambda\:h}{2\left(\lambda\:h{c}_{1}-{c}_{2}\right)};\:{C}_{m}\left({x}_{m}\right)=1$$

$$\:{C}_{m}^{{\prime\:}}\left({x}_{m-1}\right)=\frac{\lambda\:\left({c}_{1}-1\right)}{2\left(\lambda\:h{c}_{1}-{c}_{2}\right)};{C}_{m}^{{\prime\:}}\left({x}_{m+1}\right)=\frac{\lambda\:\left(1-{c}_{1}\right)}{2\left(\lambda\:h{c}_{1}-{c}_{2}\right)};{C}_{m}^{{\prime\:}}\left({x}_{m}\right)=0$$

$$\:{C{\prime\:}}_{m}^{{\prime\:}}\left({x}_{m-1}\right)={C{\prime\:}}_{m}^{{\prime\:}}\left({x}_{m+1}\right)=\frac{{\lambda\:}^{2}{c}_{2}}{2\left(\lambda\:h{c}_{1}-{c}_{2}\right)};{C{\prime\:}}_{m}^{{\prime\:}}\left({x}_{m}\right)=\frac{{-\lambda\:}^{2}{c}_{2}}{\left(\lambda\:h{c}_{1}-{c}_{2}\right)}$$

When utilizing exponential cubic B-spline as the basis functions in the fundamental DQM Eq. (2.1), the resultant equation is as follows:

$$\:\frac{{\partial\:}^{\left(r\right)}{C}_{m}\left({x}_{i}\right)}{{\partial\:x}^{\left(r\right)}}={\sum\:}_{j=m-2}^{m+2}{p}_{i,j}^{\left(r\right)}{C}_{m}\left({x}_{j}\right),\:\:\:\:\:m=-\text{1,0},\dots\:,N+2,\:\:\:\:\:i=\text{1,2},\dots\:,N.\:\:$$

2.3

Utilizing exponential cubic B-spline leads to the appearance of two additional points on both the left and right sides, resulting in a total of four additional points. A modified form of the basis functions eliminates these extra points. The modified exponential cubic B-splines can be calculated as shown below at the mesh points [31]:

$$\:{G}_{1}\left(x\right)={C}_{1}\left(x\right)+2{C}_{0}\left(x\right),$$

$$\:{G}_{2}\left(x\right)={C}_{2}\left(x\right)-{C}_{0}\left(x\right),$$

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{G}_{k}\left(x\right)={C}_{k}\left(x\right)for\:\:\:k=\text{3,4},\dots\:,N-2,$$

2.4

$$\:{G}_{N-1}\left(x\right)={C}_{N-1}\left(x\right)-{C}_{N+1}\left(x\right),$$

$$\:{G}_{N}\left(x\right)={C}_{N}\left(x\right)+2{C}_{N+1}\left(x\right),$$

When r = 1 in Eq. (2.1), the resulting equation is:

$$\:{G}_{k}^{{\prime\:}}\left({x}_{i}\right)=\:{\sum\:}_{j=1}^{N}{p}_{i,j}^{\left(1\right)}{G}_{k}\left({x}_{j}\right),\:\:for\:\:\:\:i=\text{1,2},\dots\:,N,\:\:\:\:k=\text{1,2},\dots\:,N.$$

2.5

$\:A{\overrightarrow{p}}^{\left(1\right)}\left[i\right]=\overrightarrow{T}\left[i\right]$ $\:for\:\:\:\:i=\text{1,2},\dots\:,N.$ (2.6)

Using the specified MATLAB software, the proposed approach allows for the solution of this system and the determination of the weighting coefficient values. By substituting DQM-based approximations of spatial derivatives with exponential B-spline basis functions, which can convert the system into an ordinary differential equation (ODE). This resulting ODE can be solved using various numerical methods. In this work, the resultant ODE system has been numerically solved using a strong stability-preserving time-stepping Runge-Kutta (SSP-RK43) technique [41].

Once the optimal parameter value is determined using the LOOCV to minimize error, numerical solutions can be generated within predefined spatial domains and time intervals. This approach combines the reliability of DQM with exponential B-spline basis functions and the stability of the SSP-RK43 technique, providing accurate solutions for complex differential equations.

2.3 Leave-One-Out Cross-Validation (LOOCV) algorithm

The LOOCV method provides a way to evaluate how well a machine learning algorithm will perform when it is used to make predictions. Here’s how the LOOCV procedure works:

The dataset is divided into $\:n$ subsets, where $\:n$ is the total number of observations in the dataset.
In each iteration, one observation is set aside as the test set, and the remaining $\:n-1$ observations are used to train the model.
The model is then tested on the single observation that was left out, and this process is repeated $\:n$ times, ensuring each observation serves as a test set once.
The performance of the model is evaluated by calculating the average error rate across all iterations.

In this context, the LOOCV is also used to compare the exact and numerical solutions for problems with specified initial conditions, allowing for error minimization. After selecting the parameter value that minimizes error, we can generate numerical values within specific domains and time intervals. This approach enhances accuracy and reliability by using a comprehensive evaluation method, leveraging all available data.

2.4 Pseudo-code of the Numerical Scheme

The exponential modified cubic B-spline differential quadrature technique with LOOCV is also available in the literature for finding numerical solutions of nonlinear Schrodinger equation [30] and nonlinear Sine-Gorden equation [34]. The pseudo-code for the numerical scheme using the differential quadrature method (DQM) with exponential cubic B-spline basis functions and LOOCV optimization for the parameter $\:\lambda\:$ involved in the basis functions is shown here:

Initialize the parameter $\:\lambda\:$ for the exponential cubic B-spline basis functions.
Define the grid points and the corresponding weights for the DQM.
Define the differential equations to be solved.
Implement the LOOCV algorithm:
- Loop over each data point in the dataset.
- For each data point:

1) Remove the data point from the dataset.

2) Solve the differential equations using DQM with the updated dataset.

3) Compute the error between the numerical solution and the observed data.

4) Update the parameter $\:\lambda\:$ to minimize the error using optimization techniques like gradient descent or genetic algorithms.

5) Repeat steps (1)-(4) for all data points.

Once the optimal parameter $\:\lambda\:$ is determined than proceed to solve the differential equations using DQM with the entire dataset.
Output the numerical solution obtained.

3.1 Error norms formulas

The effectiveness of the proposed methodologies and their comparison has been verified through numerical solutions of four problems of the FHN equation, with the absolute, $\:{L}_{2}$, $\:{L}_{\infty\:\:}$, and $\:RMS$ error norms formulas used to compute numerical errors are:

$$\:\text{A}\text{b}\text{s}\text{o}\text{l}\text{u}\text{t}\text{e}\:\text{e}\text{r}\text{r}\text{o}\text{r}=\left|{u}_{exact}\left({x}_{i,}t\right)-{u}_{numerical}\left({x}_{i,}t\right)\right|$$

3.1

$$\:{L}_{2}=\sqrt{h{\sum\:}_{i=1}^{N}{\left|{u}_{exact}\left({x}_{i,}t\right)-{u}_{numerical}\left({x}_{i,}t\right)\right|}^{2},}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

3.2

$$\:{L}_{\infty\:}=\text{max}\left(\left|{u}_{exact}\left({x}_{i,}t\right)-{u}_{numerical}\left({x}_{i,}t\right)\right|\right),$$

3.3

$$\:RMS=\sqrt{\frac{1}{\text{N}{\sum\:}_{\text{i}=1}^{\text{N}}{\left|{u}_{exact}\left({x}_{i,}t\right)-{u}_{numerical}\left({x}_{i,}t\right)\right|}^{2},}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

3.4

where $\:{u}_{exact}$ and $\:{u}_{numerical}$ represent the exact and numerical solutions respectively and $\:\text{N}$ represents the number of partitions of the domain.

Example 3.1

The one-dimensional nonlinear general FHN Eq. (1.1) in the domain with the exact solution [43]:

$$\:u\left(x,t\right)=\frac{1}{2}+\frac{1}{2}\text{tanh}\left(\frac{1}{2\sqrt{2}}\left(x-\frac{2\zeta\:-1}{\sqrt{2}}t\right)\right);\:\:\:\:\:\:\:x\in\:\left[-10,\:10\right]\:\&\:\text{t}\ge\:0$$

3.5

with boundary conditions derived from the exact solution and the initial conditions:

$$\:u\left(x,0\right)=\frac{1}{2}+\frac{1}{2}\text{tanh}\left(\frac{x}{2\sqrt{2}}\right);\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(3.6\right)\:\:\:\:\:$$

A numerical solution of the general FN equation is obtained for $\:a=-10$, $\:b=10$,$\:\:\varDelta\:t=0.001$ and $\:\zeta\:=0.75\:$for $\:N=51$ at various time levels. The $\:{L}_{2}$, $\:{L}_{\infty\:}$, and $\:RMS$ errors are calculated in Table 3.1 for time intervals 0.2, 0.5, 1, 1.5, 2, 3, 4, 5 and 7 with the parameter $\:\lambda\:=0.014729$ calculated using LOOCV. The obtained results are comparable or even better accuracy to those available in Jiwari et al. [16]. The absolute errors with $\:N=51$ at the different time levels with respect to the knot point is also showcased in Table 3.2 that confirms that the obtained numerical results are comparable with the available exact solutions. Figures 3.1 presents the comparison of numerical and exact solutions of the FN equation at different time intervals. Figure 3.2 illustrates the surface plots of the solutions with 51 nodes to discuss the physical behaviour of the equation along with time.

Table 3.1

Comparisons of errors for Example 3.1 to validate the scheme with errors reported in literature.
	Present				Jiwari et al. [16]
t	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$	$\:\varvec{R}\varvec{M}\varvec{S}$	CPU (seconds)	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$	$\:\varvec{R}\varvec{M}\varvec{S}$
0.2	2.5659e-09	4.3470e-05	4.9344e-11	0.495236	2.3012e-06	4.7416e-05	1.5880e-05
0.5	1.6643e-08	1.1288e-04	3.2006e-10	0.553982	5.5695e-06	1.2312e-04	3.8433e-05
1.0	7.2281e-08	2.4076e-04	1.3900e-09	0.904211	1.1864e-05	2.6261e-04	8.1870e-05
1.5	1.7988e-07	3.8563e-04	3.4593e-09	1.026189	1.9400e-05	4.2096e-04	1.3387e-04
2.0	3.5916e-07	5.4974e-04	6.9069e-09	1.303757	2.8162e-05	5.9999e-04	1.9433e-04
3.0 4.0	1.0518e-06 2.4936e-06	9.4619e-04 1.4548e-03	2.0227e-08 4.7954e-08	1.681855 2.132800	4.9735e-05 -	1.0324e-03 -	3.4320e-04 -
5.0 7.0	5.2666e-06 1.9383e-05	2.1071e-03 4.0152e-03	1.0128e-07 3.7275e-07	2.541709 3.732292	1.1395e-04 -	2.3050e-03 -	7.8638e-04 -

Table 3.2

Absolute errors at different knot points for Example 3.1 at different time-levels.
x	t = 0.5	t = 1	t = 2	t = 3
-8	1.0745e-06	8.1342e-06	3.1685e-05	3.1685e-05
-6	1.5577e-09	1.0555e-07	1.8740e-06	1.8740e-06
-4	1.2725e-07	1.7532e-07	2.1339e-07	2.1339e-07
-2	1.6552e-07	2.5014e-07	2.0657e-07	2.0657e-07
0	1.2564e-07	3.5339e-07	7.9368e-07	7.9368e-07
2	3.2215e-07	7.3645e-07	1.4380e-06	1.4380e-06
4	1.6774e-07	2.8375e-07	1.7138e-07	1.7138e-07
6	7.6015e-10	1.1653e-07	3.9629e-06	3.9629e-06
8	1.1810e-06	1.1381e-05	6.1993e-05	6.1993e-05

Example 3.2

Consider Newell-whitehead equation i.e., special type of FHN Eq. (1.1) with. Its analytic solution considered as [39]:

$$\:u\left(x,t\right)=\frac{1}{2}+\frac{1}{2}\text{tanh}\left(\frac{-1}{2\sqrt{2}}\left(x-\frac{3}{\sqrt{2}}t\right)\right);\:\:x\in\:\left[-10,\:10\right]\:\&\:\text{t}\ge\:0$$

3.7

With initial conditions as follows:

$$\:u\left(x,0\right)=\frac{1}{2}+\frac{1}{2}\text{tanh}\left(\frac{-x}{2\sqrt{2}}\right);$$

3.8

and boundary conditions taken from the exact solution.

A numerical solution of the general FN equation is obtained for $\:a=-10$, $\:b=10$, $\:\varDelta\:t=0.001,\:N=21$ and $\:\zeta\:=-1\:$at various time levels. The results are compared with earlier findings available in the literature. The $\:{L}_{2}$, $\:{L}_{\infty\:}$, and $\:RMS$ errors are calculated in Table 3.3 for time $\:t=0.001,\:0.002,\:\text{a}\text{n}\text{d}\:0.003$ with the parameter $\:\lambda\:=0.999934$ calculated using LOOCV. The obtained results are comparable to those available in Arora and Bhatia [39]. The absolute errors with $\:N=51$ at the different time levels with respect to the knot point is also showcased in Table 3.4 that confirms that the obtained numerical results are comparable with the available exact solutions. Figures 3.3 presents the comparison of numerical and exact solutions of the FN equation at different time intervals for Example 3.2 with $\:N=21$. Figure 3.4 illustrates the surface plots of the solutions with 21 nodes to discuss the physical behaviour of the equation along with time.

Table 3.3

Comparisons of errors for Example 3.2 to validate the scheme with errors reported in literature.
	Present					Arora and Bhatia [39]
t	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$		$\:\varvec{R}\varvec{M}\varvec{S}$	CPU (seconds)	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$	$\:\varvec{R}\varvec{M}\varvec{S}$
0.001	2.2883e-12		1.2728e-06	1.0402e-13	0.213532	2.5076e-08	8.7088e-09	1.9004e-09
0.002	9.1534e-12		2.5475e-06	4.1606e-13	0.236610	2.3994e-08	6.7952e-08	5.2359e-09
0.003	2.0595e-11		3.8241e-06	9.3615e-13	0.240576	1.2648e-07	4.5460e-08	9.9202e-09

Table 3.4

Absolute errors of Example 3.2 with $\:\varvec{N}=21$, $\:\varDelta\:\varvec{t}=0.001$ and $\:\varvec{t}=\text{1,2},\:3.$
x	t = 1	t = 2	t = 3
-8	2.4363e-05	4.7621e-05	5.4469e-05
-6	1.7059e-06	2.4807e-06	3.0830e-06
-4	4.4370e-06	1.6514e-06	6.4182e-07
-2	9.7642e-07	2.7366e-06	1.4593e-06
0	4.4810e-05	1.1090e-07	4.0350e-06
2	8.1468e-05	1.1101e-05	9.5080e-06
4	2.1570e-05	1.1271e-04	8.0603e-05
6	2.0480e-05	2.1129e-04	1.7288e-03
8	2.0010e-04	2.4849e-03	1.4491e-02

Example 3.3

Consider the FN equation domain [-20, 20] for the values of which reduces to the Fisher’s equation [44]. Along with boundary condition for the exact solution as:

$$\:u(x,t)\:=\frac{1}{1+\text{e}\text{x}\text{p}\left(\frac{x}{\sqrt{2}}-\frac{t}{2}\right)}$$

3.9

The solution of the equation is obtained with the proposed approach with the initial condition

$$\:u(x,0)\:=\frac{1}{1+\text{e}\text{x}\text{p}\left(\frac{x}{\sqrt{2}}\right)}$$

3.10

The numerical solution of the equation is obtained for different time levels for time $\:t\le\:5\:$with $\:N=251\:$for $\:\varDelta\:t=0.01$. The $\:{L}_{2}$, $\:{L}_{\infty\:}$, and $\:RMS$ errors are calculated in Table 3.5 for time $\:t=0.001,\:0.002,\:\text{a}\text{n}\text{d}\:0.003$ with the parameter $\:\lambda\:=0.999934$ calculated using LOOCV. The obtained results are comparable to those available in Singh et al. [44]. The absolute errors with $\:N=21$ at the different time levels with respect to the knot point is also showcased in Table 3.6. As shown by the obtained results, the numerical solutions are in good agreement with the exact solution. Figures 3.5 presents the comparison of numerical and exact solutions of the FN equation at different time levels. The surface plot of the solutions with 251 nodes to discuss the physical behavior of the equation along with time is presented in Fig. 3.6.

Table 3.5

Comparisons of errors of Example 3.3 to validate the scheme with errors reported in literature.
	Present				Singh et al. [44]
t	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$	RMS	CPU (seconds)	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$
0.5	1.2825e-12	9.2624e-07	5.0891e-15	3.705071	1.21e-07	1.66e-07
1.0	2.1720e-12	1.1893e-06	8.6190e-15	3.707353	3.05e-07	4.01e-07
1.5	3.6842e-12	1.5271e-06	1.4620e-14	4.043707	5.49e-07	7.06e-07
2.0	6.1851e-12	1.9608e-06	2.4544e-14	4.333491	8.69e-07	1.10e-06
2.5	1.0309e-11	2.5178e-06	4.0909e-14	4.514205	1.29e-06	1.61e-06
3.0	1.7110e-11	3.2329e-06	6.7898e-14	4.801520	1.84e-06	2.26e-06

Table 3.6

Comparison of absolute errors of Example 3.3 with $\:\varvec{N}=21$, $\:\varDelta\:\varvec{t}=0.01$ and $\:\varvec{t}=\text{1,2},\:3.$
x	t = 1	t = 2	t = 3
-8	6.3397e-05	6.1443e-05	4.2354e-05
-6	1.8122e-06	3.5541e-06	4.8202e-06
-4	9.7119e-06	7.1353e-06	4.5746e-06
-2	2.8717e-05	1.8571e-05	5.1006e-06
0	5.3160e-05	8.4761e-05	6.2940e-05
2	1.0378e-04	1.6624e-04	1.2294e-04
4	1.2309e-05	3.0096e-05	1.4479e-04
6	4.5727e-06	3.3730e-05	1.3958e-04
8	1.7227e-04	4.5610e-04	8.6448e-04

Example 3.4

Consider the equation with the initial and boundary condition taken from the exact solution [45] for the equation defined as:

$$\:u\left(x,t\right)=\:1-\frac{\text{cosh}\left(\frac{t}{2}-\frac{x}{\sqrt{2}}\right)+\text{sinh}\left(\frac{t}{2}-\frac{x}{\sqrt{2}}\right)}{2+\text{cosh}\left(\frac{t}{2}-\frac{x}{\sqrt{2}}\right)+\text{sinh}\left(\frac{t}{2}-\frac{x}{\sqrt{2}}\right)}$$

3.11

The equation is solved on domain [-10,10] using present approach and the errors are presented in Table 3.6 for different time levels on time $\:t\le\:1$. The obtained results are obtained for $\:\zeta\:=1\:$with $\:N=21$ and $\:\varDelta\:t=0.0001$. The $\:{L}_{2}$, $\:{L}_{\infty\:}$, and $\:RMS$ errors are calculated in Table 3.7 for various time levels with the parameter $\:\lambda\:=0.999934$ calculated using LOOCV. The absolute errors with $\:N=21$ at the different time levels with respect to the knot point is also showcased in Table 3.8. As shown by the obtained results, the numerical solutions are in good agreement with the exact solution. Figures 3.7 presents the comparison of numerical and exact solutions of the FN equation at $\:t=\text{1,3},5,\:and\:7$. The surface plot of the solutions with 21 nodes to discuss the physical behavior of the equation along with time is presented in Fig. 3.8.

Table 3.7

Comparisons of errors of Example 3.4 to validate the scheme with errors reported in literature.
t	$\:{\varvec{L}}_{2}$	$\:{\varvec{L}}_{\varvec{\infty\:}}$	RMS	CPU (seconds)
0.001	3.1886e-07	2.0534e-07	1.4494e-08	0.287100
0.002	6.1904e-07	4.1032e-07	2.8138e-08	0.303760
0.005	1.5308e-06	1.0231e-06	6.9582e-08	0.338395
0.01	3.0452e-06	2.0372e-06	1.3842e-07	0.503110
0.02	6.0382e-06	4.0387e-06	2.7446e-07	0.753933
0.05	1.4731e-05	9.8347e-06	6.6957e-07	1.218910
0.1	2.8330e-05	1.8828e-05	1.2877e-06	2.033497
0.2	5.2656e-05	3.4509e-05	2.3935e-06	3.710440
0.5	1.0893e-04	6.5943e-05	4.9513e-06	8.704131
1.0	1.6990e-04	1.0485e-04	7.7225e-06	16.798947
2.0	2.4098e-04	1.6489e-04	1.0954e-05	32.898018
5.0	3.9359e-04	2.8011e-04	1.7890e-05	82.676119

Table 3.8

Absolute errors of Example 3.4 with $\:\varvec{N}=21$, $\:\varDelta\:\varvec{t}=0.0001$ and $\:\varvec{t}=\text{1,2},\:3.$
x	t = 1	t = 2	t = 3
-8	2.9536e-06	2.7026e-06	1.9718e-06
-6	7.0746e-06	6.3235e-06	4.4160e-06
-4	3.7276e-06	2.9322e-06	4.1584e-06
-2	7.9109e-05	6.1238e-05	2.8677e-05
0	7.8273e-05	1.8736e-05	4.0112e-05
2	3.0532e-05	1.3058e-04	2.1843e-04
4	1.6311e-05	1.8881e-05	1.6612e-05
6	5.4415e-06	1.5401e-05	2.5803e-05
8	1.9747e-06	5.2168e-06	1.1024e-05

This study presents an innovative numerical technique called the “exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) with Leave-one-out cross-validation (LOOCV)” to find numerical solutions for the Fitzhugh-Nagumo equation. The LOOCV approach is used to optimize the parameter $\:\lambda\:$, a key component of the exponential cubic B-spline basis functions, leading to improved accuracy and stability in the results. This combination offers a novel and highly effective approach to solving the Fitzhugh-Nagumo equation. The approach’s validity is confirmed through various numerical tests, which show that this method delivers solutions that closely match exact results. The results demonstrate that this approach not only provides high accuracy but also is simpler to implement compared to existing methods in the literature. This work encourages further research into other basis functions requiring optimized parameters, such as hyperbolic tension B-splines, and suggests exploring additional cross-validation techniques to refine the algorithm’s performance. The versatility of this algorithm opens possibilities for applying it to more complex problems in science and engineering. Overall, this study establishes a robust framework for solving nonlinear partial differential equations, demonstrating significant potential for further applications and development in the field.

Author contribution statement

All authors listed have significantly contributed to the development and the writing of this article.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.

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No competing interests reported.

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Numerical simulation of soliton solutions of nonlinear Fitzhugh-Nagumo equation by using LOOCV with exponential B-spline with Significant Applications in Neurosciences

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Abstract

Figures

1. Introduction

1.1 Applications of the Fitzhugh-Nagumo Model in Neurosciences

1.1.1 Understanding Neuronal Excitability

1.1.2. Analysis of Action Potential Dynamics

1.1.3. Wave Propagation in Neural Tissues

1.1.4. Pattern Formation and Oscillations

1.1.5. Modeling Cardiac Dynamics

1.1.6. Exploring Neurophysiological Disorders

1.1.7. Synaptic Interactions and Network Dynamics

1.1 Leave-One-Out Cross-Validation (LOOCV)

2. Numerical Scheme

2.1 The Differential Quadrature Method with B-spline Basis Functions Algorithm

2.2 Exponential modified cubic B-spline differential quadrature method

2.3 Leave-One-Out Cross-Validation (LOOCV) algorithm

2.4 Pseudo-code of the Numerical Scheme

3. Numerical Applications

3.1 Error norms formulas

4. Conclusion

Declarations

References

Additional Declarations

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	Present				Jiwari et al. [16]
t	\(\:{\varvec{L}}_{2}\)	\(\:{\varvec{L}}_{\varvec{\infty\:}}\)	\(\:\varvec{R}\varvec{M}\varvec{S}\)	CPU (seconds)	\(\:{\varvec{L}}_{2}\)	\(\:{\varvec{L}}_{\varvec{\infty\:}}\)	\(\:\varvec{R}\varvec{M}\varvec{S}\)
0.2	2.5659e-09	4.3470e-05	4.9344e-11	0.495236	2.3012e-06	4.7416e-05	1.5880e-05
0.5	1.6643e-08	1.1288e-04	3.2006e-10	0.553982	5.5695e-06	1.2312e-04	3.8433e-05
1.0	7.2281e-08	2.4076e-04	1.3900e-09	0.904211	1.1864e-05	2.6261e-04	8.1870e-05
1.5	1.7988e-07	3.8563e-04	3.4593e-09	1.026189	1.9400e-05	4.2096e-04	1.3387e-04
2.0	3.5916e-07	5.4974e-04	6.9069e-09	1.303757	2.8162e-05	5.9999e-04	1.9433e-04
3.0 4.0	1.0518e-06 2.4936e-06	9.4619e-04 1.4548e-03	2.0227e-08 4.7954e-08	1.681855 2.132800	4.9735e-05 -	1.0324e-03 -	3.4320e-04 -
5.0 7.0	5.2666e-06 1.9383e-05	2.1071e-03 4.0152e-03	1.0128e-07 3.7275e-07	2.541709 3.732292	1.1395e-04 -	2.3050e-03 -	7.8638e-04 -