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Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains.

Abstract In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and ( 2 − γ ) $(2-\gamma )$ th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

In this paper, we establish a mathematical model with two delays to reflect the intrinsic and extrinsic incubation periods of virus in dengue transmission. The basic reproduction number $ R_0 $ of the model is defined. It is proved that the disease-free equilibrium is stable when $ R_0 \lt 1 $ and the positive equilibrium is stable when $ R_0 \gt 1 $ . Next, we derive an estimation formula for the reproduction number $ R_t $ when the human population is partially susceptible to dengue. As an application, the $ R_t $ values of dengue transmission in Singapore in the years 2013–2015 are estimated. Our estimation method can be applied to estimating $ R_t $ of other infectious diseases, especially when the human population is not completely susceptible to the disease.

In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order α. The main idea involves the generalized linear interpolation and so we term the numerical scheme the gL1 scheme. The stability and convergence of the numerical scheme are analyzed using the energy method. It is proven that the temporal convergence order is (2−α) for a general temporal mesh. Simulation is carried out to verify the efficiency of the proposed numerical scheme.

Abstract In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.

We consider the following complementary Lidstone boundary value problem $$\begin{array}{c}(-1)^{m}y^{(2m+1)}(t)= F(t,y(t), y'(t)),~~t\in[0,1]\\ y(0)=0, y^{(2k-1)}(0)=y^{(2k-1)}(1)=0, 1\leq k\leq m. \end{array}$$ The nonlinear term $F$ depends on $y'$ and this derivative dependence is seldom investigated in the literature. Using a variety of fixed point theorems, we establish the existence of one or more positive solutions for the boundary value problem. Examples are also included to illustrate the results obtained.

By applying Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three bounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature.

We consider two systems of Volterra integral equations $$ u_i(t)=h_i(t) + int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),dots, u_n(s))ds, quad 1leq ileq n $$ where t is in the closed interval $[0,T]$, or in the half-open interval $[0,T)$. By an argument originated from Brezis and Browder [8], criteria are offered for the existence of solutions of the systems of Volterra integral equations. We further establish the existence of constant-sign solutions, which include positive solutions (the usual consideration) as a special case. Some examples are also presented to illustrate the results obtained.

We consider the system of Volterra integral equations $$ \begin{array}{l} u_i(t)=\int_{0}^{t}g_i(t,s)[P_i(s,u_1(s),u_2(s),\cdots, u_n(s)) + Q_i(s,u_1(s),u_2(s),\cdots, u_n(s))]ds, t\in [0,T],1\leq i\leq n \end{array} $$ where $T>0$ is fixed and the nonlinearities $P_i(t,u_1,u_2,\cdots,u_n)$ can be singular at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$ Criteria are offered for the existence of fixed-sign solutions $(u_1^*,u_2^*,\cdots,u_n^*)$ to the system of Volterra integral equations, i.e., $\theta_iu_i^*(t)\geq 0$ for $t\in [0,1]$ and $1\leq i\leq n,$ where $\theta_i\in \{1,-1\}$ is fixed. We also include an example to illustrate the usefulness of the results obtained.

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation $$ \left\{ \begin{array}{lll} (\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t,x)|≤μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be |f(t,x)|≤μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption |f(t,x)|≤μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.

We study the existence and uniqueness of positive solutions for a class of singular m-point boundary value problems of second order differential equations on a measure chain. A sharper sufficient condition for the existence and uniqueness of Crd⁡1[0,T] positive solutions as well as Crd⁡1[0,T] positive solutions is obtained by the technique of lower and upper solutions and the maximal principle theorem.

The purpose of this paper is to obtain some sufficient conditions for the global existence of multiple positive periodic solutions of a delayed stage-structured plant-hare model with a toxin-determined functional response. Some novel estimation techniques to construct two open subsets for a priori bounds are employed.

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model with M-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

Three-point boundary value problems of second-order differential equation with a p-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.

We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial P2m(t) of degree 2m, which involves interpolating data at the odd-order derivatives. For P2m(t) we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a (2m+1)th order differential equation and the complementary Lidstone boundary conditions.