Fisher Equation Research Articles

A Regional Catastrophe Bond Pricing Model and Its Application in Indonesia’s Provinces

The national scale of catastrophic losses risk linked to state catastrophe bonds (SCB) is enormous. It can reduce investors’ interest in buying them because the capital required and the loss probability are also significant. To overcome this, the SCB can be made on a smaller regional scale, known as a regional catastrophe bond (RCB). Through RCBs, the catastrophic loss risk investors bear becomes smaller, which can increase investors’ interest in buying them. Unfortunately, RCB issuance faced a fundamental obstacle, where its complex pricing model needed further study. Therefore, this study aims to model it. The model uniquely involves the inflation rate modeled using the Fisher equation and the nonbinary scheme of coupon and redemption value payments modeled by a compound Poisson process. In addition, the model is applied to Indonesia’s catastrophe data, resulting in all provinces’ RCB price estimation and the effects of several variables on RCB price. This research can guide the RCB pricing process of the country’s regions. The estimated RCB prices can be used by Indonesia’s government if RCBs are to be issued one day. Finally, the effects of the inflation rate, catastrophe intensity, and geographical location on RCB prices can guide investors in selecting bond portfolios.

A Hybrid Non-Polynomial Spline Method and Conformable Fractional Continuity Equation

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using (L2 and L∞) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods’ unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors.

Symmetry Analysis and Wave Solutions of the Fisher Equation Using Conformal Fractional Derivatives

In the present article, the time fractional Fisher equation is considered in conformal form to study the application of the Lie classical method and quantitative analysis. The Lie symmetry method has been applied to find the infinitesimal generators and symmetry reductions of the fractional Fisher equation. The obtained reduced form of the equation is solved by the method of G ′ / G , which gives different forms of solutions. The theory of bifurcation has been utilized in the reduced form to check the stability and nature of critical points by transforming the equations into an autonomous system. Some phase portraits have been drawn at different critical points by the use of maple.

Conservation Laws and Exact Solutions for Time-Delayed Burgers–Fisher Equations

A generalization of the time-delayed Burgers–Fisher equation is studied. This partial differential equation appears in many physical and biological problems describing the interaction between reaction, diffusion, and convection. New travelling wave solutions are obtained. The solutions are derived in a systematic way by applying the multi-reduction method to the symmetry-invariant conservation laws. The translation-invariant conservation law yields a first integral, which is a first-order Chini equation. Under certain conditions on the coefficients of the equation, the Chini type equation obtained can be solved, yielding travelling wave solutions expressed in terms of the Lerch transcendent function. For a special case, the first integral becomes a Riccati equation, whose solutions are given in terms of Bessel functions, and for a special case of the parameters, the solutions are given in terms of exponential, trigonometric, and hyperbolic functions. Furthermore, a complete classification of the zeroth-order local conservation laws is obtained. To the best of our knowledge, our results include new solutions that have not been previously reported in the literature.

Newfangled Linearization Formula of Certain Nonsymmetric Jacobi Polynomials: Numerical Treatment of Nonlinear Fisher’s Equation

This article is devoted to deriving a new linearization formula of a class for Jacobi polynomials that generalizes the third-kind Chebyshev polynomials class. In fact, this new linearization formula generalizes some existing ones in the literature. The derivation of this formula is based on employing a new moment formula of this class of polynomials and after that using suitable symbolic computation to reduce the resulting linearization coefficients into simplified forms that do not contain any hypergeometric functions or sums. The new formula is employed along with some other formulas and with the utilization of the spectral tau method to obtain numerical solutions to the nonlinear Fisher equation. The presented method is used to convert the equation governed by its underlying conditions into a nonlinear system of equations. The solution of the resulting system can be obtained through any suitable standard numerical scheme. To demonstrate the efficiency and usefulness of the proposed algorithm, some examples are shown, including comparisons with some existing techniques in the literature.

Exact solutions of time fractional generalized Burgers–Fisher equation using exp and exponential rational function methods

Exact solutions of time fractional generalized Burgers–Fisher equation using exp and exponential rational function methods

Exact and Approximate Solutions for Linear and Nonlinear Partial Differential Equations via Laplace Residual Power Series Method

The Laplace residual power series method was introduced as an effective technique for finding exact and approximate series solutions to various kinds of differential equations. In this context, we utilize the Laplace residual power series method to generate analytic solutions to various kinds of partial differential equations. Then, by resorting to the above-mentioned technique, we derive certain solutions to different types of linear and nonlinear partial differential equations, including wave equations, nonhomogeneous space telegraph equations, water wave partial differential equations, Klein–Gordon partial differential equations, Fisher equations, and a few others. Moreover, we numerically examine several results by investing some graphs and tables and comparing our results with the exact solutions of some nominated differential equations to display the new approach’s reliability, capability, and efficiency.

Quantized-Output-Based Exponential Stabilization of Fuzzy DPSs With Time-Varying Delay and Nonlinearities

The exponential stabilization of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. With the employment of the Takagi-Sugeno (T-S) fuzzy model, the nonlinear DPSs are described as a fuzzy parabolic partial differential equation (PDE). Then, with the consideration of the measurement output and control input information quantized by a kind of dynamic quantizers, fuzzy static and dynamic output feedback controllers are proposed. The time-delayed feedback controllers are also provided in the forms of static and dynamic. On the basis of the derived T-S fuzzy PDE model, Lyapunov-based control design strategies are proposed with respect to the effect of quantization. Moreover, by constructing appropriate Lyapunov functionals, sufficient conditions for designing the output feedback control gains and the quantizers' adjusting parameters which guarantee the exponential stability of the fuzzy closed-loop system are provided in terms of standard linear matrix inequalities (LMIs). Finally, we explore the output feedback controllers into the FitzHugh-Nagumo (FHN) equation and Fisher equation as applications. The efficiency of the design strategies is indicated through the simulation results.

Globally Stable Difference Schemes for the Fisher Equation

Globally Stable Difference Schemes for the Fisher Equation

Optimal convergence rate of the explicit Euler method for convection–diffusion equations II: High dimensional cases

AbstractThis is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection–diffusion equations [Zhang et al. Appl. Math. Lett. 131 (2022), 108048] which focuses on high‐dimensional linear/nonlinear cases under Dirichlet/Neumann boundary conditions. Several new difference schemes are proposed based on the explicit Euler discretization in temporal derivative and centered difference discretization in spatial derivatives. The priori estimate of the improved difference scheme with application to the constant convection coefficients is performed under the maximum norm and the optimal convergence rate four is achieved when the step‐ratios along each direction equal to . Also we give partial results for the three‐dimensional case. The improved difference schemes have essentially improved the CFL condition and the numerical accuracy comparing with the classical difference schemes. Numerical examples involving two‐/three‐dimensional linear/nonlinear problems under Dirichlet/Neumann boundary conditions such as the Fisher equation, the Chafee–Infante equation and the Burgers' equation substantiate the good properties claimed for the improved difference scheme.

A high‐order numerical technique for generalized time‐fractional Fisher's equation

The generalized time‐fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high‐order numerical scheme for the generalized time‐fractional Fisher's equation. The time‐fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of , where and are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem.

On a finite population variation of the Fisher–KPP equation

In this paper, we formulate a finite population variation of the Fisher–KPP equation using the fact that the reaction term can be generated from the replicator dynamic using a two-player two-strategy skew-symmetric game. We use prior results from Ablowitz and Zeppetella to show that the resulting system of partial differential equations admits a travelling wave solution, and that there are closed form solutions for this travelling wave. Interestingly, the closed form solution is constructed from a sign-reversal of the known closed form solution of the classic Fisher equation. We also construct a closed form solution approximation for the corresponding equilibrium problem on a finite interval with Dirichlet and Neumann boundary conditions. Two conjectures on these corresponding equilibrium problems are presented and analysed numerically.

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

Series-form solutions of generalized fractional-fisher models with uncertainties using hybrid approach in Caputo sense

The field of fuzzy calculus has emerged as a powerful mathematical tool which can effectively deal with uncertainties and impressions that are common in real-world situations. In particular, it has proven useful in modeling and analysis of complex biological systems with uncertain parameters. The current study focuses on analysis of (n+1)-dimensional fractional Fisher equations (FFEs) in fuzzy environment. The objective is to provide semi-analytical solutions for fuzzy (n+1)- dimensional FFEs by considering Caputo-gH fractional derivative. The uncertainty in initial conditions is injected through triangular fuzzy numbers and obtained fuzzy (n+1)-dimensional FFEs are solved using hybrid of homotopy perturbation with Laplace transform in fuzzy-Caputo sense, which provides a powerful mathematical framework for examining complex behavior. The derived series solutions are validated against existing results from the literature and found to be improved. The obtained results are analyzed by means of determining the fuzzy solutions and residual errors at varying fractional orders, membership function, spatial coordinate x, and time t. These analytical findings are visualized in graphical form for ease of comprehension. The conducted study yields significant insights about the behavior of fractional model having uncertain conditions, and highlights the efficiency of proposed methodology. The results of this study have important implications for understanding the dynamics of biological systems with uncertainty, and hence can be useful in wide variety of applications in different fields such as ecology, epidemiology, and economics.

A new numerical scheme non-polynomial spline for solving generalized time fractional Fisher equation

In this paper, a novel numerical scheme is developed using a new construct by non-polynomial spline for solving the time fractional Generalize Fisher equation. The proposed models represent bacteria, epidemics, Brownian motion, kinetics of chemicals and fuzzy systems. The basic concept of the new approach is constructing a non-polynomial spline with different non-polynomial trigonometric and exponential functions to solve fractional differential equations. The investigated method is demonstrated theoretically to be unconditionally stable. Furthermore, the truncation error is analyzed to determine the or-der of convergence of the proposed technique. The presented method was tested in some examples and compared graphically with analytical solutions for showing the applicability and effectiveness of the developed numerical scheme. In addition, the present method is compared by norm error with the cubic B-spline method to validate the efficiency and accuracy of the presented algorithm. The outcome of the study reveals that the developed construct is suitable and reliable for solving nonlinear fractional differential equations.

A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis

Burger and Fisher diffusion transfer properties and reactions from the characteristics are studied using a non-linear equation. The nonlinear fractional Burgers–Fisher equation (NFB-FE) appears in realistic physical situations such as ultra-slow kinetics, Brownian motion of particles, anomalous diffusion, polymerases of ribonucleic acid and deoxyribonucleic acid, continuous random movement, and formation of wave patterns. The present study focuses on the collocation scheme based on the shifted Chebyshev basis (SCB) and the compact finite difference method to obtain the numerical scheme of the NFB-FE. The simulation model is created in the two steps: Initially, a semi-discrete is formed in a temporal sense, applying a linear approximation with an accuracy order of two. Next, we examine the unconditional stability and the convergence order. In the second stage, the collocation approach based on the SCB of the fourth type is used to discretize the spatial derivative parts and generate the full-discrete scheme.

On Fisher's equation with the fractionalp‐Laplacian

In the present note, we show that if the initial date is between zero and one, then the global solution of a Dirichlet Fisher‐KPP equation with the fractional ‐Laplacian is also between zero and one. As a consequence, a large‐time behavior of the global solution is discussed. In addition, some blow‐up results including a general Lipschitz nonlinearity are given. The proofs mainly rely on special symmetry and invariance properties of the fractional ‐Laplacian.

Importance of Addressing Structural Break While Estimating Fisher Effect Hypothesis: A Time Series Analysis of Bangladesh

Fisher Equation, asserted by Irving Fisher in his celebrated book ‘The Theory of Interest’, reveals that nominal interest rate adjusts with the inflation rate at the same rate in order to keep real interest rate constant. Using yearly time series data from Bangladesh for the period 1987 – 2020, this paper tested the validity of the Fisher effect considering the variables- Nominal Interest Rate (Advance Rate), Real Interest Rate and Inflation Rate. With a view to rationalizing the existence of structural break, Clemente - Montanes – Reyes (1998) unit root test was performed to identify the integration order of the variables of interest. After controlling structural break, Gregory- Hansen (1996) cointegration test applied for detecting the long run relationship by adopting ARDL framework. As the results revealed, Partial Fisher effect prevails for Bangladesh in short run. More specifically, keeping the real interest constant the impact of inflation rate on nominal interest rate is 0.16 percent with a lag effect in both ARDL model but parameters are stable only when we address structural break in the model. Thus, establishing that empirical identification of short run relationship among variables and existence of Fisher effect is subject to proper consideration of structural break. Social Science Review, Vol. 38(2), December 2021 Page 175-196

Traveling Wave of Biological Invasion: Effects of Swapping Migration

The problem of biological invasion is an issue of increasing significance in conservation ecology. Fisher equation is one of the traditional methods to describe biological invasion. In ecosystems, however, species usually live in patchy environments. In this study, a modified Fisher equation is derived, where the recently developed “swapping migration” is applied and the relation between invasion velocity and patch capacity is discussed.

Study of nonlinear generalized Fisher equation under fractional fuzzy concept

<abstract><p>Fractional calculus can provide an accurate model of many dynamical systems, which leads to a set of partial differential equations (PDE). Fisher's equation is one of these PDEs. This article focuses on a new method that is used for the analytical solution of Fuzzy nonlinear time fractional generalized Fisher's equation (FNLTFGFE) with a source term. While the uncertainty is considered in the initial condition, the proposed technique supports the process of the solution commencing from the parametric form (double parametric form) of a fuzzy number. Next, a joint mechanism of natural transform (NT) coupled with Adomian decomposition method (ADM) is utilized, and the nonlinear term is calculated through ADM. The obtained solution of the unknown function is written in infinite series form. It has been observed that the solution obtained is rapid and accurate. The result proved that this method is more efficient and less time-consuming in comparison with all other methods. Three examples are presented to show the efficiency of the proposed techniques. The result shows that uncertainty plays an important role in analytical sense. i.e., as the uncertainty decreases, the solution approaches a classical solution. Hence, this method makes a very useful contribution towards the solution of the fuzzy nonlinear time fractional generalized Fisher's equation. Moreover, the matlab (2015) software has been used to draw the graphs.</p></abstract>