Fractals

This paper describes the required and adequate conditions for controllability and optimal controls of Atangana–Baleanu–Caputo (ABC) neutral fractional integrodifferential equations (NFIE) with noninstantaneous impulses. Measure of noncompactness, semigroup theory, fractional calculus and fixed point techniques act as the main tools in deriving the results. An illustration is offered to enhance our theoretical outcomes.

This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional order which has been introduced by Atangana, Baleanu and Caputo (ABC). The aforesaid type problems have numerous applications in fluid mechanics and hydrodynamics to model various problems of flow phenomenons. We establish some sufficient conditions for the existence and uniqueness of solution to the proposed problem by using classical fixed point results due to Banach and Krasnoselskii. Further, on using tools of the nonlinear analysis, sufficient conditions are developed for Hyers–Ulam (HU) type stability results. A pertinent example is given to justify our results.

Currently, the entire planet is suffering from a contagious epidemic infection, 2019-nCOV due to newly detected coronavirus. This is a lethal infectious virus that has destroyed thousands of lives all over the world. The important aim of this study is to investigate a susceptible-infected-treatment-recovered (SITR) model of coronavirus (2019-nCOV) with bi-modal virus spread in a susceptible population. The considered 2019-nCOV model is analyzed by two fractional derivatives: the Caputo and Atangana–Baleanu–Caputo (ABC). For the Caputo model, we present a few basic mathematical characteristics such as existence, positivity, boundedness and stability result for disease-free equilibria. The fixed-point principle is used to establish the existence and uniqueness conditions for the ABC model solution. We employed the Adams–Bashforth–Moulton (ABM) numerical technique for the Caputo model solution and the Toufik–Atangana (TA) numerical approach for the ABC model solution. Finally, using MATLAB, the simulation results are shown to highlight the impact of arbitrarily chosen fractional-order and model parameters on infection dynamics.

This paper is devoted to establish some theoretical and computational results for a coupled system of drug therapy process. The considered problem is investigated by using the concept of piecewise modeling. We apply piecewise differentiation and integration to study the proposed model. We derive some adequate results for existence and uniqueness for solution to the proposed problems by using fixed point results. Further, we apply some numerical approach based on Newton interpolation formula for equally spaced arguments to find numerical results corresponding to our proposed model. We use some real values and two sets of fractional order to present the dynamics of the considered model graphically.

The application of the recently proposed integral and differential operators known as the fractal-fractional derivatives and integrals has opened doors to ongoing research in different fields of science, engineering, and technology. These operators are a convolution of the fractal derivative with the generalized Mittag-Leffler function with Delta-Dirac property, the power law, and the exponential decay law with Delta-Dirac property. In this paper, we aim to extend the work in the literature by applying these operators to a modified stretch–twist–fold (STF) flow based on the STF flow related to the motion of particles in fluids that naturally occur in the dynamo theorem. We want to capture the dynamical behavior of the modified STF flow under these operators. We will present the numerical schemes that can be used to solve these nonlinear systems of differential equations. We will also consider numerical simulations for different values of fractional order and fractal dimension.

This research work is devoted to endeavor some results for a delay implicit impulsive type problem under Atangana–Baleanu fractional derivative. The concerned derivative utilizes a nonlocal and non-singular kernel. We build some hypotheses to prove our results. We use Banach and Krasnoselskii fixed point theorems to derive the required results. We consider the following problem involving nonlocal and non-singular fractional derivative with delay term:

The COVID-19 pandemic touched about 200 countries of the globe. A strategy is given in this paper by considering a seven-compartment mathematical model with the inclusion of asymptomatic and symptomatic populations with waning immunity under the piecewise derivative concept of singular and nonsingular kernels, respectively. We investigate the dynamics of COVID-19 with the new framework of piecewise fractional derivative in the sense of Caputo and Atangana–Baleanu–Caputo fractional operators. The said analysis includes at least one solution and unique solution analysis with piecewise derivative in two subintervals. The proposed model is carried out by the approximate solution of piecewise numerical iterative technique of Newton polynomial. Each equation is written separately for the algorithm of numerical technique. Graphical representation for the proposed piecewise derivable model has been simulated with the available data at various global orders lying between 0 and 1 for both the subintervals. Such type of analysis will be very good and helpful for all those global problems where sudden or abrupt variation occurs.

A dynamical model of SARS-CoV-2 in fractional derivative using the cases of coronavirus of the fourth wave is presented. We construct basically the model in an integer case, and later it is extended to a fractional-order system by applying the Atangana–Baleanu operator definition. We give some background definitions and results for the fractional-order model. We present for the disease-free case that the model is locally asymptotically stable when ${{ℛ}}_0<1$. The global dynamics of the fractional model are given when ${{ℛ}}_0\le 1$ for the disease-free case. The model is further extended to fractional stochastic piecewise equations in the Atangana–Baleanu case. The reported cases from the fourth wave in Pakistan starting from July 1 up to November 16, 2021 are considered for the estimation of the parameters. We fitted our model to the suggested data and obtained the numerical value of the basic reproduction number ${{ℛ}}_0\approx 0.9775$ for fractional order. We give the data fitting to both the fractional and piecewise stochastic differential equations, and show them both as having a good fitting to the data. We use further the numerical values of the model parameters and present its numerical results graphically using the effective numerical approaches. Some sensitive parameters that are reasonable for disease eliminations are used to obtain the graphical results.

The aim is to study the dynamics of Coronavirus model using stochastic methods. Threshold parameter $R_0$ is obtained for the model. Afterwards, both the disease-free equilibrium (DFE) and endemic equilibrium (EE) points are acquired and the stability of the model is discussed. Both the equilibrium points are locally asymptotically stable. Euler–Maruyama, stochastic Euler scheme (SES), stochastic fourth-order Runge–Kutta scheme (SRKS) and stochastic non-standard finite difference technique (SNFDT) are applied to solve the model equations. Euler–Maruyama, SES, SRKS fail for large time step size, while, SNFDT preserves the dynamics of the proposed model for any step size. Numerical comparison of applied methods is provided using different step sizes.

We apply the Ritz method to approximate the solution of optimal control problems through the use of polynomials. The constraints of the problem take the form of differential equations of fractional order accompanied by the boundary and initial conditions. The ultimate goal of the algorithm is to set up a system of equations whose number matches the unknowns. Computing the unknowns enables us to approximate the solution of the objective function in the form of polynomials.

A system of three nonlinear equations that depict chaotic behavior was considered in this work. We looked at four cases: one with a power-law property; one with a fading memory effect; one with a transition from stretched exponential to power-law; and one with a stochastic component. We have presented a detailed analysis of stability and we have suggested a number similar to the reproductive number in the case of epidemiology. At this stage, we did not provide a physical meaning for such a number. However, such a number may be useful. A numerical scheme based on the Lagrange polynomial interpolation was used to solve the respective models numerically. Numerical simulations were performed for different values of fractional orders and densities of randomness and piecewise ideas.

The purpose of this paper is to establish some sufficient conditions needed for the existence and uniqueness of solutions to the coupled system of fractional hybrid differential equations (FHDEs). We make use of the prior estimate method to support the main results. An example is also provided to justify our main results.

The use of mathematical modeling in the exploration of epidemiological disorders has increased dramatically. Mathematical models can be used to forecast how viral infections spread, as well as to depict the likely outcome of an outbreak and to support public health measures. In this paper, we present useful ideas for finding existence of solutions of the novel coronavirus 2019-nCoV/ SARS-CoV-2 model via fractional derivatives by using fuzzy mappings. Three classes of fractional operators were considered including Atangana–Baleanu, Caputo–Fabrizio and Caputo. For each case, we introduce the fuzzination in the study of the existence of a system of solutions. A fresh numerical scheme was proposed for each scenario, and then numerical simulations involving various parameters of Atangana–Baleanu fractional-order were shown utilizing numerical solutions.

The adaptation of fractional calculus (FC) in biological mathematical model takes the research in the area of the public health to a new level. The fractional definitions and related mathematical tools have had a significant impact on biological models analysis. The main goal of this paper is to examine the dynamical behavior of a predator–prey model under Caputo derivative. We analyze some special results such as convergence analysis, stability and operational matrix for the proposed Caputo model. For solution of the model, we present a new numerical technique-based Laguerre wavelet. In addition, we graphically compare the numerical results obtained using Laguerre wavelets and Lagrange polynomial interpolation.

In this paper, we use newly introduced piecewise classical-global and classical-fractional operators to study the dynamics of the Leptospirosis disease model. The existence and uniqueness of the solution to piecewise derivatives are examined for the suggested problem. The piecewise iterative Newton polynomial method is used to obtain an approximate solution to the suggested problem. In addition, a numerical scheme for the piecewise Leptospirosis model with integer and fractional orders is established. The numerical simulation of the piecewise derivable problem under consideration is presented in classical as well as various fractional orders.

Atangana and Araz have introduced a new term called piecewise differentiation and integration which has been successfully applied to epidemiology in these days. This concept paves the way of seeing different patterns of modeling real-world problems. In this study, a mathematical model for the transmission dynamics of Contagious Bovine Pleuropneumonia (CBPP) is considered and modified based on vaccination and antibiotic therapy by using these operators. This model is classified into six classes namely susceptible, exposed, vaccinated, infectious, continuously infected, and treated classes. The model is solved by a numerical scheme based on Newton interpolation polynomial. The existence and uniqueness of the model are investigated. Various patterns of piecewise concepts have been presented through the numerical figures.

We apply the noncompactness measure by Mawhin to establish sufficient conditions for the existence and uniqueness (EU) of solution to a nonlinear Cauchy problem. The aforesaid problem is investigated under impulsive and nonlinear integral boundary conditions. Further, the problem under consideration contains Caputo-type fractional-order derivative while two different kinds of delay are also involved. After establishing the existence results, we derive some adequate results for the stability analysis of Hyers–Ulam (HU) type. Here, it should be kept in mind that the proposed tools can reduce the strong compact condition by weaker compactness. For justification, we provide some examples also.

In this paper, we introduce a new class of fractional impulsive systems of functions with respect to another function in which the order of the fractional derivative and the kernel function is associated with the impulses. We derive the solution representation, investigate the existence, and uniqueness of solutions of such a Caputo-type fractional impulsive system. Besides, the data dependence of the system is discussed. Our arguments are based on some classical fixed-point styles. Three various examples are provided to illustrate the validation of the main results. An open problem is presented in the conclusion section to bring the attention to a more general setting.

In this paper, we study the blow-up problem for Fujita-type equations with a general absorption term $f$ on finite graphs and locally finite graphs, respectively. We prove that if $f$ satisfies appropriate conditions, then the solutions of the equation blow up in finite time. The obtained results are generalization of those given by the author in previous papers. At the end of the paper, we provide some numerical experiments to illustrate the applicability of the obtained results.

The COVID-19 pandemic started, a global effort to develop vaccines and make them available to the public, has prompted a turning point in the history of vaccine development. In this study, we formulate a stochastic COVID-19 epidemic mathematical model with a vaccination effect. First, we present the model equilibria and basic reproduction number. To indicate that our stochastic model is well-posed, we prove the existence and uniqueness of a positive solution at the beginning. The sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. For controlling the transmission of the disease by the application of external sources, the theory of stochastic optimality is established. The nonlinear least-squares procedure is utilized to parametrize the model from actual cases reported in Pakistan. The numerical simulations are carried out to demonstrate the analytical results.

This study is concerned with the existence and uniqueness results for fuzzy differential equations (FDEs). The existence result of FDEs in the space of linear correlated (LC) fuzzy number with LC-differentiability is the goal of this work. Linear FDEs of first order in the space of LC-fuzzy number mostly have many solutions and sometimes have no solutions. Existence theory insures the existence of unique solution of FDEs. Therefore, we propose to study the existence results of LC-fuzzy solution of FDEs of first order. The existence criteria for the unique LC-fuzzy solution for both symmetric and nonsymmetric basic fuzzy number will be studied in this work. Some examples are given to show the usability of the criteria for a unique LC-fuzzy solution. The plots of the fuzzy solutions of examples are also provided.

Social media networking sites, such as Instagram, Facebook, and Twitter, have become an inextricable part of our everyday lives. These social media networks are useful for sharing news, images, and other information. The study of social media sentiment has recently received a lot of attention, especially in Arabic sentiment analysis. Social media sites are distinguished by unusual language that differs from the traditional format of the language. As a result, there is a necessity for efficient ways for analyzing the massive amount of new word variants that appear regularly in the digital world and online world. This study proposes a piecewise Stochastic Gradient Descent (SGD)-based model for sentiment classification. The TF-IDF-based term weighting scheme is employed for textual feature representation. For enhancing the model performance, stemming and partially ordered microword representation of tweets with varying look ahead distances is employed. Also, various n-gram models are considered for textual feature representation, which also improves the model performance. The proposed model is simulated and evaluated with the help of publicly available tweet corpus⁹ which is a balanced tweet corpus. The effectiveness of the proposed model is estimated using various performance evaluation metrics. According to the experimental observations, the proposed method accurately categorizes the testing and validation tweets set with an accuracy of up to 92.23%.

In this paper, we aim to analyze the effect of treatment and protection measures on the spread of infectious disease. Therefore, we formulate a delayed mathematical which contains a discrete delay which stands for the duration of protection and distributed delay that represents the repulsed individuals from treatment. The main result is to show that the model has threshold behavior governed by ${\Re }_0$. For ${\Re }_0<1$, it is shown that the disease-free equilibrium is globally stable. However, for ${\Re }_0>1$, it is proved that the semiflow is uniformly persistent, and the endemic equilibrium is globally stable. Another purpose is fixed in this paper, which consists of studying the effect of treatment and protection measures on the spread of the disease, and to distinguish the minimal protection and treatment measure for reducing the basic reproduction number below one. The results are supported numerically using numerical simulations.

Smoking has many harmful effects due to its toxic chemicals, which cause serious diseases. Our major interest in this research is to formulate a spatiotemporal mathematical model that predicts the evolution of smoking in society using a mathematical model that includes the large mobility of individuals. Indeed, we are concerned to study the global asymptotic stability of the unique positive steady state related to this model. The principal objective of this paper is to show that the smoking has no threshold dynamics as it has been shown for a large sample of epidemic models, and the consumption of cigarettes is always persistent, and based on the assumption of the parameters that verify the Lipschitz condition. We proved that the investigated model is always persistent and the unique positive equilibrium state is always globally stable. The mathematical finding is supported numerically using numerical simulations.

This research work is related to establish a powerful algorithm for the computation of numerical solution to nonlinear variable order integro-differential equations (VO-IDEs). The adopted procedure is based on the Haar Wavelet Method (HWM) to compute the required numerical solution to the proposed problem. Further, in the considered problem, a proportional-type delay term is involved, which is also known as the pantograph equation. For a physical problem to investigate the computational purposes, we need to first ensure its existence. For this purpose, we utilize classical fixed results given by Banach and Schauder to establish the sufficient conditions for existence of at least one approximate solution to the proposed problem. Two pertinent examples are given, where the error analysis is also recorded.

The propagation attributes of waves and its modeling maneuvers have a significant role in maritime, coastal engineering, and ocean. In the geographical fields, waves are primary source of environmental process owed to energy conveyance on the floating structure or on the synthetic field. This study aims to investigate the new auxiliary equation method to obtain analytical solutions of the nonlinear Hirota model with fractional order. The fractional model is developed by utilizing Riemann–Liouville, $𝔅$, and the fractional-order Atangana–Baleanu differential operator in Riemann–Liouville sense. The solitonic patterns of the nonlinear fractional Hirota equation successfully surveyed, where the exact solutions are presented by rational, trigonometric, hyperbolic, and exponential functions. The contravene of surveyed results with the substantially recognized result is executed which states the novelty of obtained results. Three dimensional as well as two-dimensional comparison is presented for a couple of Hirota model solutions which are revealed diagrammatically for appropriate parameters by using Mathematica. We strongly believe that this study will help physicists to predict some new conceptions in the field of mathematical physics.

Speech is the primary method of communication among humans; it is the main form of communication to deliver emotions and thoughts. Intuitively, humans would find it convenient to communicate with machines through speech. The automatic speech recognition model’s primary goal is to transcribe or identify the word sequence represented by the acoustic signals or speech. In this advanced era of digital media and e-learning platforms, the effective use of voice recognition technology to deliver a more individualized and effective educational experience in e-learning initiatives. It helps the students to improve their oral pronunciation skills. Even though Arabic is one of the most spoken languages, the research works related to Arabic speech and text are lacking when compared to other languages. Here, we propose a hybrid model of $K$-Nearest Neighbor (KNN) classifier and Dynamic Time Warping (DTW) for implementing a noise-resilient speech recognition system for isolated words in the Arabic language. The model is implemented with Mel-Frequency Cepstral Coefficients (MFCC) and its piecewise first and second derivatives as feature representation models. The proposed model is implemented in Python and simulated using Arabic Speech Corpus for Isolated Words [A. Alalshekmubarak and L. Smith, On improving the classification capability of reservoir computing for Arabic speech recognition, in International Conference on Artificial Neural Networks (Springer, Cham, 2014), pp. 225–232]. The proposed model is implemented with (1) MFCC alone for feature representation, (2) combination of MFCC with Delta coefficients, and (3) combination of MFCC with Delta and Delta–Delta coefficients. The implemented model is evaluated using different test sets of varying sizes of 100, 200, 500, 1000, 1500, and 2000. Evaluation is performed for both noised and noiseless speech using these three feature representation models and performed an evaluation of these three models. For evaluating this model’s performance in noised conditions, both white and babble noises of various signal-to-noise ratio values such as 10 dB, 20 dB, and 30 dB are added into the noiseless speech and estimate the classification accuracy in those situations. The proposed KNN–DTW model with MFCC outperformed all the other models in the literature. MFCC with Delta and Delta–Delta coefficients is effective than other two models using MFCC and MFCC with Delta coefficients. The proposed model could be used to recognize the isolated word recognition of grade 1 textbook vocabulary in classroom noisy environment for providing a better interactive classroom environment to Kuwaiti elementary students to enhance their math learning level. In an interactive computer-based education environment for Mathematics learning, the proposed model could be recognized as the spoken utterances of students.

Fractal structures are everywhere around us as they occur naturally or are artificially simulated. Applied mostly in engineering domains that include neural networks, fractal processes help fostering architectural design. For instance, fractal models are commonly used to design new machine learning algorithms for neural networks. Differential operators that can artificially trigger such fractal processes become a valuable asset for engineers. We use in this paper the fractal derivative combined to the fractional dynamic to analyze the chaotic proto-Lü system. That combined operator, known as the fractal-fractional derivative (FFD), is relatively new in the literature and has many features still to be discovered. The piecewise model of the proto-Lü system combining the fractal-fractional and classical derivatives is analyzed and solved numerically. In the study, we start by providing a succinct summary of fundamentals behind the FFD and equations of the proto-Lü system. The latter comprise different models with $n$ scrolls each ($n\in \mathbb{N}$) corresponding to its nth cover. We focus on applying the FFD on the third cover of the proto-Lü system that is solved numerically via the Haar wavelets. Numerical simulations show the maintenance of the multiscroll chaotic attractors. The representations for the piecewise model also show the maintenance of the triple cover both in a stretched form and a self-similarity process. Additionally, we observe the capacity of those attractors to perform self-replication in a fractal structure that varies with the fractional-order derivative.

In recent years, scholars have studied the chaotic behavior in the fractional dynamic systems and found that the fractional dynamic systems have unique properties that the integer dynamic systems do not have. Therefore, the numerical simulation of fractional chaotic systems is very important. This paper introduces a high-precision numerical method for the fractional-order Rössler chaotic systems. Complex dynamic behavior of the fractional-order Rössler chaotic systems is shown by using the present method. We observe some novel dynamic behaviors in numerical simulation which are unlike any that have been previously found in numerical experiments or theoretical studies. The simulation results of numerical experiments demonstrate the effectiveness of the present method.

Herein, a spectral scheme is implemented and analyzed for numerically handling general Volterra–Fredholm integral equations (VFIEs), for this purpose, the linearly independent Fibonacci polynomials are utilized as basis functions for the solution, then the spectral collocation process is used to transform the integral equation into a system of algebraic equations with undetermined coefficients. The error, convergence and stability analyses of the scheme are discussed in-depth, some numerical examples are exhibited to ensure the applicability, efficiency and accuracy of the solver.

In this paper, we investigated some essential provisions for the existence and stability of the solution to integral boundary value problems with proportional delay of fractional order Atangana–Baleanu–Caputo (ABC) derivative. By the guidance of fixed point theory, we acquire the deserted results. Moreover, different types of Ullam–Hyers stabilities are investigated for the proposed problem. We also provide an appropriate example for illustrative purposes.

Among the family of multi-scroll chaotic attractors, we have the multi-folded torus chaotic attractor. Concerns were raised among applied scientists on how to control, merge, split or modify the whole or parts of these multi-folded torus attractors in their implementation processes while conserving the same circuit diagram and the same system of differential equations. In this paper, we strive to address these concerns by using the piecewise process, that is, applying to the modified multi-folded torus system, the piecewise derivative made of the fractal-fractional and classical derivatives. A modified piecewise linear odd function is also used in the process. The modified model is first solved numerically thanks to the wavelet method and numerical simulations are performed. The results show the progressive transformation of the initial multi-folded torus attractors into stretched or reduced multi-folded torus attractors. However, the chaotic state of the system is preserved by the piecewise modification.

Many approaches have been proposed in recent decades to represent the behaviors of certain complicated global problems appearing in a variety of academic domains. One of these issues is the multi-step behavior that some situations exhibit. Abdon and Seda devised new operators known as “piecewise operators” to deal with such problems. This paper presents the dynamics of the tumor–immune–vitamins model in the sense of a piecewise derivative. The piecewise operator considered here is composed of classical and Caputo operators. The existence and uniqueness of the solution with a piecewise derivative are presented with the aid of fixed point results. With the help of the Newton polynomial, a numerical scheme is presented for the examined model. The attained results are visualized through simulations for different fractional orders.

We aim in this paper to study the effect of harvesting on predator–prey interaction in the case of prey herd behavior using a fractional-order model. Herd behavior has a crucial role in the surviving of species where it gave them the sufficient protection for the prey that perform it. The objective of using the fractional-order model is to model the memory effect measured by the order of the fractional derivative on the mutual interactions. Further, we aim to seek the effect of inner competition among the predators (also super-predators) on the evolution of the three species. For the mathematical results, we will show the local stability of the equilibria, and show the effect of memory rate and harvesting of the asymptotic behavior of the solution. Moreover, an efficient numerical scheme has been used to provide the numerical illustrations for our study.

Prüfer transformation is more effective and flexible in studying the spectral analysis of boundary value problem than using the classical methods in operator theory. The goal of this paper is to study Prüfer approach to spectral analysis of periodic Sturm–Liouville problem with transmission condition. Since we are dealing with a singular problem, the characteristic function we obtained is a piecewise function. At the end of the study, the existence of eigenvalues of investigated problem by using Prüfer transformation is given.

In this paper, the therapeutic interactions are discussed with the aid of a mathematical model and piecewise differentiation technique. The model is based on the interaction between cell populations in therapeutic stress and cell populations in neoplastic transformation, referring specifically to triple-negative breast cancer (TNBC). The proposed computational approach provides an opportunity for the qualitative and parametric analysis of the clinical trials in question. The proposed mathematical model is based on the analysis of cell–cell interactions according to a logic that examines pathological stress and its influence on the dynamics of the cell population involved in tumor pathology. Detailed literature review and dynamical analysis of the proposed hypothesis are provided in this paper. The existence and non-negativity of the solutions are exploited, the hypothesis is stabilized; it is then further demonstrated with the aid of the piecewise derivative and the relevant application of the formula of Newton interpolation.

The aim of this paper is to study the dynamics of Dengue disease model using a novel piecewise derivative approach in the sense of singular and non-singular kernels. The singular kernel operator is in the sense of Caputo, whereas the non-singular kernel operator is the Atangana–Baleanu Caputo operator. The existence and uniqueness of a solution with piecewise derivative are examined for the aforementioned problem. The suggested problem’s approximate solution was found using the piecewise numerical iterative Newton polynomial approach. In terms of singular and non-singular kernels, a numerical scheme for piecewise derivatives has been established. The numerical simulation for the piecewise derivable problem under consideration is drawn using data for various fractional orders. This work makes the idea of piecewise derivatives and the dynamics of the crossover problem clearer.

The purpose of this work is to study the memory effect analysis of Caputo–Fabrizio time fractional diffusion equation by means of cubic B-spline functions. The Caputo–Fabrizio interpretation of fractional derivative involves a non-singular kernel that permits to describe some class of material heterogeneities and the effect of memory more effectively. The proposed numerical technique relies on finite difference approach and cubic B-spline functions for discretization along temporal and spatial grids, respectively. To ensure that the error does not amplify during computational process, stability analysis is performed. The described algorithm is second-order convergent along time and space directions. The computational competence of the scheme is tested through some numerical examples. The results reveal that the current scheme is reasonably efficient and reliable to be used for solving the subject problem.