by of Commons permits In this article, we propose a new wavelet method for solving nonlinear Katugampola fractional differential equations on an arbitrary interval. We have introduced a new wavelet, which we named as the Katugampola Gegenbauer wavelet (KGW), and constructed its new operational matrices of Katugampola fractional integrations as well as Katugampola fractional derivatives. The Katugampola Gegenbauer wavelet and its operational matrices are combined with the Adomian polynomials to propose a new method for the solution of nonlinear Katugampola fractional differential equations. The purpose of using the Adomian polynomials is to handle the nonlinearities in the equations. Furthermore, we have provided a detailed methodology for implementing the proposed approach to nonlinear Katugampola fractional differential equations. A detailed error analysis is also performed for the proposed method. The proposed method is implemented on several nonlinear Katugampola fractional differential equations to show the reliability, efficiency, and accuracy of the method.

The primary objective of this paper is to develop an efficient numerical method for solving two-dimensional nonlinear fractional partial differential equations in both temporal and spatial domains. The proposed approach combines a fast algorithm with Shifted Legendre wavelets to efficiently handle fractional derivatives in space and time. The resulting scheme is straightforward to implement and computationally efficient. An interpolation technique is employed to convert the nonlinear problems into equivalent linear forms, after which the proposed method is applied. Convergence analysis confirms that the numerical solutions are in good agreement with the corresponding analytical solutions. Numerical results, presented in both tabular and graphical formats, further demonstrate the accuracy and efficiency of the proposed method. In addition, comparisons with existing methods from the literature are provided to highlight the superior performance of the proposed approach.

This study presents a comprehensive analysis of a discrete-time predator-prey model, uniquely combining rigorous theoretical investigation with empirical validation using four decades of moose-wolf interaction data. The proposed model incorporates a Ricker-type growth function for the prey population, which inherently ensures the positivity of solutions, a crucial aspect of ecological realism. Our qualitative analysis identifies biologically feasible equilibrium points and examines their local stability to delineate conditions for species coexistence or extinction. The rich and bifurcating dynamical behavior of the model is studied by using center manifold theory and normal forms to investigate codimension-one bifurcations, specifically transcritical and period-doubling bifurcations. Furthermore, the first Lyapunov exponent is explicitly derived to characterize Neimark-Sacker bifurcations emerging from the positive fixed point. The research also explores complex codimension-two bifurcations, including fold-flip and strong resonances (1:2, 1:3, and 1:4), revealing intricate pathways between stability, periodic oscillations, and chaotic regimes. To bridge theoretical control strategies with practical wildlife management, the Ott-Grebogi-Yorke (OGY) method is applied to mitigate chaotic dynamics, offering ecological interpretations for the necessary perturbations. This work thereby provides a robust framework for understanding and managing complex ecological systems through the synergy of mathematical modeling and long-term empirical data. Numerical simulation is provided to illustrate the theoretical discussion.MSC2020 Classification : 37N25, 92D40, 39A30, 37G15, 37G10

Purpose This paper aims to develop a numerical method for the numerical solutions of nonlinear Caputo fractional differential equations. Design/methodology/approach We introduced fractional-order discrete Hahn polynomials (FDHPs) by modifying classical discrete Hahn polynomials (DHPs) and used them to generate a wavelet named fractional-order discrete Hahn wavelet (FDHW). Findings The solution is approached by approximating the appropriate terms in nonlinear Caputo fractional differential equations using FDHWs and converting the problem to a system of algebraic equations. We have expanded nonlinear terms by using Taylor’s series to linearize the nonlinear Caputo fractional differential equations. Graphical results and numerical simulations are provided, and a comparison with some well-known methods in the literature is made to illustrate the reliability and accuracy of the proposed wavelet technique. Originality/value Many engineers or scientists can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

This study introduces a new wavelet framework, referred to as the tempered fractional Gegenbauer wavelet (TFGW), for the numerical solution of tempered variable-order differential equations. We construct the TFGW operational matrices for tempered variable-order integration and develop the TFGW method to efficiently solve Caputo-tempered variable-order ordinary and boundary value problems. To further enhance computational efficiency for partial differential equations involving both time-fractional and spatial variable-order derivatives, we propose the L1-TFGW method based on the L1 approximation and the fast TFGW method, which incorporates a fast algorithm for fractional time derivatives in combination with the TFGW approach for spatial operators. For nonlinear problems, a fast-quasi TFGW method is devised by coupling quasilinearization with the fast TFGW strategy. The orthonormality of TFGW is established, and corresponding operational matrices are derived, including those tailored for boundary value problems. Error analyses are provided, and extensive numerical simulations demonstrate the accuracy, efficiency, and robustness of the proposed methods. The results confirm that the TFGW-based techniques offer a reliable and effective computational framework for linear and nonlinear Caputo-tempered variable-order models. To the best of our knowledge, this is the first work to introduce such wavelet-based fast algorithms for tempered fractional and spatial variable-order differential equations, providing a valuable tool for scientists and engineers dealing with complex multiscale fractional dynamics.

PurposeThe objectives of this study are threefold: (1) to introduce the tempered Chebyshev wavelet (TCW), (2) to propose the TCW method for solving linear tempered Caputo fractional diffusion-type equations and (3) to develop a fast tempered Chebyshev wavelet (fTCW) method for solving one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations.Design/methodology/approachThe fTCW method integrates the TCW method with L2-1 sigma and sum-of-exponentials approximations of the tempered Caputo fractional time derivative. For this purpose, we introduce the TCW and derive its new operational matrices for tempered fractional integration by using the hypergeometric function. For nonlinear problems, we employ the interpolation technique in conjunction with operational matrices and fast approximations. The efficiency of the fTCW method is demonstrated through comparisons with the exact solution and the solution obtained using the TCW method.FindingsWe have derived the TCW operational matrix of tempered fractional integration and the TCW operational matrix of tempered fractional integration for boundary value problems. These matrices, in conjunction with the fast approximations of the tempered Caputo fractional time derivative and the interpolation technique, form the basis for the construction of the fTCW method. We present a detailed methodology for solving linear tempered Caputo fractional equations using the TCW method. Additionally, we provide a comprehensive methodology for solving both one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations using the fTCW method. The convergence and error analysis of both methods are thoroughly discussed. Numerical simulations are presented to validate and illustrate the theoretical results. These simulations demonstrate the effectiveness and accuracy of the proposed methods by solving three test problems, including one linear and two nonlinear tempered Caputo fractional diffusion-type equations. The results are compared with analytical solutions or with each other to highlight the efficiency and precision of the TCW and fTCW methods.Originality/valueMany engineers and scientists can utilize the presented methods for solving their linear and nonlinear tempered Caputo fractional diffusion-type models.

PurposeThe objective of this study is to introduce a more efficient method than classical wavelet methods, for solving linear and nonlinear Caputo fractional variable-order diffusion-type equations.Design/methodology/approachWe first construct the Haar wavelet operational matrices for Caputo variable-order integration. Next, we integrate these matrices with the L2 - 1 sigma approximations to solve linear Caputo fractional variable-order diffusion-type equations. For nonlinear problems, we employ the quasilinearization technique in conjunction with the operational matrices and L2 - 1 sigma approximations. The proposed method is named "the modified Haar wavelet (mHw) method." The efficiency of the mHw method is demonstrated through comparisons with the exact solution and the solution obtained using the classical Haar wavelet method.FindingsWe have derived the Haar wavelet operational matrix of variable-order integration and the variable-order integration matrix of Haar wavelet for boundary value problems. These matrices, in conjunction with the L2 - 1 sigma approximations and the quasilinearization technique, form the basis for the construction of the mHw method. We also provide the theoretical analysis of the mHw method. Additionally, the mHw method is shown to be second-order accurate in both time and space domains. We also performed the comparison with the classical Haar wavelet method. Numerical simulations are presented to validate and illustrate the theoretical results.Originality/valueMany engineers and scientists can utilize the presented method for solving their linear and nonlinear Caputo fractional variable-order models.

This paper introduces a numerical approach by generalizing Legendre wavelets for solving nonlinear Caputo–Hadamard fractional differential equations. The methodology involves the extension of classical Legendre wavelets, namely the generalized Legendre wavelets (gLWs), along with the development of operational matrices for Hadamard fractional integration and Caputo–Hadamard fractional differentiation. The proposed method combines the gLWs with the Adomian decomposition method to address the nonlinearities inherent in fractional equations through Adomian polynomials. A detailed methodology is presented for applying the proposed method to nonlinear Caputo–Hadamard fractional differential equations, accompanied by error analysis and numerical simulations to demonstrate its reliability and accuracy.

Climate change is expected to increase the frequency and magnitude of extreme weather events in Pakistan. During and immediately after such events many locations are inaccessible, yet the government needs evidence-based data to help plan effective responses. This paper uses spatial analysis and spectral mapping to assess the agricultural damage from the 2022 floods in Pakistan. Spatial analysis could be a vital tool in assessing and verifying damage from disaster events. The paper emphasizes the need to build spatial analysis capacities in provincial crop reporting services and the Ministry of National Food Security and Research.

Chemical reactions reveal all types of exotic behavior, that is, multistability, oscillation, chaos, or multistationarity. The mathematical framework of rate equations enables us to discuss steadystates, stability and oscillatory behavior of a chemical reaction. A planar cubic dynamical system governed by nonlinear differential equations induced by kinetic differential equations for a two-species chemical reaction is studied. It is investigated that system has unique positive steady state. Moreover, local dynamics of system is studied around its positive steady state. Existence and direction of Hopf bifurcation about positive equilibrium are carried out. In order to modify the bifurcating behavior, bifurcation control is investigated. Keeping in mind, a consistency preserving discretization for continuous chemical reaction system, a discrete counterpart is proposed, and its qualitative behavior is investigated. Numerical simulation along with bifurcation diagrams are provided to illustrate the mathematical investigations.

In this paper, we present an operational matrix method to obtain numerical solutions of ‐Caputo fractional ordinary and partial differential equations. For this purpose, a fractional version of the Taylor theorem is presented in the framework of ‐fractional calculus. The method converts the underlying ordinary or partial differential equations to systems of algebraic equations. The method is accompanied by numerical examples to verify the applicability and effectiveness of the proposed method. Further, estimates of upper bounds of error for the approximations have been derived.

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order , and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models.

Teaching and research are two essential components of every university. While some institutions are dedicated to teaching (referred to as ‘teaching university’), others favor research (‘research university’), even fewer are able to strike a balance between the two. This work evaluates the correlation between research output of Lahore’s private universities and their intra-university collaboration. The authors specifically chose private universities of Lahore (Pakistan’s second largest city) because their faculty have higher teaching load, attract less funding and draw fewer excellent students. The authors employed a data-intensive approach by collecting research profiles of faculty members of all 21 private institutions and summarized them. Results show a high degree of correlation (0.846) between the university’s research output and its intra-university collaboration. At the same time, the study also presents a grim picture showing for instance, that majority of the faculty in private institutions in Lahore do not present any scientific contribution.

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

In this article, a modification is proposed for the classical Nicholson–Bailey model. It is assumed that the modified model follows all axioms of Nicholson–Bailey model except that in every generation a fraction of the hosts have a safe refuge from attack of parasitoids. It is investigated that under this assumption the modified model has stable coexistence. Furthermore, Neimark–Sacker bifurcation is interrogated at positive steady-state of modified model by implementing the normal forms theory of bifurcation. Chaos control methods based on perturbation of parameter and state feedback strategy are implemented to escape the trajectories from bifurcating and chaotic behavior. Furthermore, numerical simulations are carried out for illustration of theoretical discussion. Finally, all theoretical discussion is illustrated by taking into account real observed field data of host–parasitoid interaction.

In this article, a wavelet method is introduced for solving Caputo–Hadamard fractional differential equations on an arbitrary interval. The proposed method is the fractional‐order generalization of sine–cosine wavelets (FGSCWs). The operational matrices of fractional‐order integration are constructed for solving initial value problem as well as boundary value problem. Furthermore, numerical solution of nonlinear Caputo–Hadamard fractional differential equation is obtained with the conjunction of proposed method with quasilinearization technique. We have constructed the FGSCW operational matrix, FGSCW operational matrix of Hadamard fractional integration of arbitrary order, and FGSCW operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. Convergence analysis of the proposed method is investigated. Numerical procedure is given for both Caputo–Hadamard initial and boundary value problems. Illustrative examples show the reliability and efficiency of the proposed method and give solution with less error.

PurposeThe purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo–Hadamard fractional differential equations on a semi‐infinite domain.Design/methodology/approachWe have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo–Hadamard fractional delay differential equations.FindingsWe have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration, and Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo–Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient.Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo–Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo–Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/valueSince fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present wavelets method has never been introduced and implemented for Caputo fractional and Caputo–Hadamard fractional differential equations.

Purpose The purpose of the present work is to propose a wavelet method for the numerical solutions of Caputo–Hadamard fractional differential equations on any arbitrary interval. Design/methodology/approach The author has modified the CAS wavelets (mCAS) and utilized it for the solution of Caputo–Hadamard fractional linear/nonlinear initial and boundary value problems. The author has derived and constructed the new operational matrices for the mCAS wavelets. Furthermore, The author has also proposed a method which is the combination of mCAS wavelets and quasilinearization technique for the solution of nonlinear Caputo–Hadamard fractional differential equations. Findings The author has proved the orthonormality of the mCAS wavelets. The author has constructed the mCAS wavelets matrix, mCAS wavelets operational matrix of Hadamard fractional integration of arbitrary order and mCAS wavelets operational matrix of Hadamard fractional integration for Caputo–Hadamard fractional boundary value problems. These operational matrices are used to make the calculations fast. Furthermore, the author works out on the error analysis for the method. The author presented the procedure of implementation for both Caputo–Hadamard fractional initial and boundary value problems. Numerical simulation is provided to illustrate the reliability and accuracy of the method. Originality/value Many scientist, physician and engineers can take the benefit of the presented method for the simulation of their linear/nonlinear Caputo–Hadamard fractional differential models. To the best of the author’s knowledge, the present work has never been proposed and implemented for linear/nonlinear Caputo–Hadamard fractional differential equations.

In this article we introduce a numerical method, named Gegenbauer wavelets method, which is derived from conventional Gegenbauer polynomials, for solving fractional initial and boundary value problems. The operational matrices are derived and utilized to reduce the linear fractional differential equation to a system of algebraic equations. We perform the convergence analysis for the Gegenbauer wavelets method.We also combine Gegenbauer wavelets operational matrix method with quasilinearization technique for solving fractional nonlinear differential equation. Quasi linearization technique is used to discretize the nonlinear fractional ordinary differential equation and then the Gegenbauer wavelet method is applied to discretized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Gegenbauer wavelet method. Numerical examples are provided to illustrate the efficiency and accuracy of the methods.