This paper proposes and validates a finite‐difference numerical model for time‐fractional multiasset Black–Scholes option‐pricing problems. The model adopts a Caputo time derivative of order α ∈ (0, 1] to capture the memory effects of asset prices. A logarithmic coordinate transformation is first applied to convert the pricing equation into a constant‐coefficient convection–diffusion–reaction equation. The temporal discretization is discretized using a weighted fully implicit L1‐scheme, and the spatial difference operator is assembled as the tensor product of one‐dimensional second‐order central‐difference operators. This formulation yields discrete systems with regular sparse structures in both two‐ and three‐asset cases, so that a single sparse LU‐factorization can be reused across all time levels. Under the assumption of smooth solutions, the global error satisfies O ( Δ τ 2− α ) + O ( h 2 ). Numerical experiments are conducted to validate the present model at two stages. Manufactured‐solution tests confirm the expected temporal convergence of order 2 − α and second‐order spatial accuracy, with convergence rates independent of dimension. Point‐wise comparisons are further performed for a one‐asset European call option against the Laplace theoretical solution and for a two‐asset exchange option against the Margrabe analytical solution, which verify the pricing accuracy for low‐asset problems. As a practical application, the proposed model is employed to price a three‐asset equally weighted basket call option. Sensitivity analysis demonstrates that the impact of the fractional order on option prices depends systematically on the moneyness of the payoff. The developed model therefore provides a deterministic, order‐preserving, and cost‐effective tool for pricing European, exchange, and multiasset basket options under market memory effects. It can be directly applied to price computation and parameter‐sensitivity analysis.
The Korteweg–de Vries (KdV) equation provides a fundamental framework for describing nonlinear dispersive wave propagation and solitary-wave dynamics. In this study, a generalized KdV model incorporating fifth-order dispersion, weak viscous dissipation, linear damping/gain, and external forcing is investigated. The Hamiltonian structure and conservation properties of the classical KdV equation are first examined, and the generalized model is then analyzed using a weak-perturbation approach. The classical KdV solitary wave is used as the leading-order solution, whereas first-order corrections associated with higher order dispersion and dissipative effects are derived. In particular, an explicit solvability condition is formulated for the fifth-order dispersive correction, and an effective damping rate governing the slow evolution of the solitary-wave amplitude is obtained. The analytical results are complemented by Fourier pseudospectral simulations combined with fourth-order Runge–Kutta time integration. The numerical experiments examine the effects of the perturbation parameters on amplitude, width, propagation speed, energy evolution, and solitary-wave interactions. For the weak perturbation regimes considered, the solitary waves retain a coherent localized structure, whereas higher order dispersion produces weak dispersive radiation and profile modification and dissipation produces gradual amplitude and energy loss. The simulations also indicate a gradual departure from the elastic collision behavior of the classical KdV system, resulting in weakly inelastic interactions. These observations become more pronounced as the perturbation strength increases. The results are therefore interpreted as regime-dependent characteristics of the generalized model within the weakly perturbed parameter ranges investigated. The study provides an analytical and computational framework for examining weakly nonconservative extensions of KdV-type solitary-wave dynamics.
This research will present a comprehensive analysis of the Erdelyi–Kober (E-K) fractional integral operator and its action on various configurations of the hypergeometric form of MacRobert E-function. Analytical solutions for the application of the E-K operator to the simple MacRobert E-function, a weighted form of the function, and the Cauchy product of two distinct E-functions will be derived, resulting in generalized Fox–Wright functions. The study further explores the fundamental mathematical properties, specifically proving the linear property and the closure property within the Fox–Wright domain. Crucially, we will define the convergence criteria required for the validity of these transformations, emphasizing the necessity of the condition Rη>−1 to ensure integral existence at the origin. Furthermore, numerical simulations and graphical comparisons are provided to validate the established analytical identities, demonstrating the stability and asymptotic convergence of the transmuted functional space. Our results will provide a unified framework for evaluating fractional integrals of complex special functions, offering significant utility for problems in mathematical physics and applied analysis involving multiple-parameter functional transmutations.
A double Sawi–Adomian decomposition framework is developed for nonlinear initial-value problems with one space and one time variable. The double transform converts the time derivative and the prescribed data into a Volterra integral equation, whereas Adomian polynomials organize the nonlinear term. In contrast with earlier transform–decomposition studies that mainly report formal series, the present formulation includes a contraction condition for existence and uniqueness, a convergence criterion for the decomposition series, and both a priori and residual-based truncation bounds. A second-order nonlinear equation and the inviscid Burgers equation are examined. Their decomposition components reproduce the Taylor expansions of tanh2ξτ and ξ/1+τ, respectively. Numerical error tables show the effect of the truncation order and also expose the finite radius of convergence in the Burgers example. Thus, the contribution is a certified double-Sawi formulation rather than a claim that changing the transform alone alters the underlying Adomian approximation.
This article develops a symmetry-based integration programme for a first-order system with two dependent variables obtained in the classification of diagonalisable hydrodynamic chains. Represented in the three-dimensional space with coordinates x,y,z, the system admits a non-trivial Lie point symmetry which reduces it to an Abel differential equation of the second kind for βr, together with an auxiliary quadrature for αr. The central observation pursued here is that the reduced differential equation itself admits a nontangent Lie point symmetry. This makes it possible to interpret the integration process as a genuine successive reduction procedure: first by the Lie point symmetry of the two-dimensional first-order system and then by a symmetry of the scalar reduced equation. On regular local branches, the resulting solutions are described by a rational first integral in the r,β-plane, by explicit parametrisations of the corresponding generic algebraic curves, and by an inverse transformation back to x,y,z. The emphasis is on the successive-reduction mechanism underlying integration of the two-dimensional first-order system.
