In this study, we propose an approximate solution based on two‐dimensional shifted Legendre polynomials to solve nonlinear stochastic partial differential equations with variable coefficients. For this purpose, we have considered a Fisher‐Kolmogorov‐Petrovsky‐Piskunov (Fisher–KPP) equation with space uniform white noise for the same. New stochastic operational matrix of integration based on shifted Legendre polynomials is generated. This operational matrix reduces the problem under study into solving a system of algebraic equations. The convergence analysis is discussed, and the error bound in
L2$$ {L}^2 $$ norm is derived and obtained as
‖‖eNfalse(t,xfalse)2≤6Kfalse(16N4false)2false(1−6Sfalse)$$ {\left\Vert {e}_N\left(t,x\right)\right\Vert}^2\le \frac{6K}{{\left(16{N}^4\right)}^2\left(1-6S\right)} $$. The theoretical analysis confirms that as the degree of the approximation polynomial is increased, the solution is on par with the exact solution. The numerical examples confirm the accuracy and the applicability of the proposed method. A comparative study is carried out with explicit 1.5 order Runge–Kutta method. The time complexity of the proposed technique is also studied.
The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.
This paper suggests using fractional Euler polynomials (FEPs) to solve the fractional diffusion-wave equation in Caputo’s sense. We present the fundamental characteristics of Euler polynomials. The method for building FEPs is discussed. By basically converting fractional partial differential equations into a system of polynomial equations, these qualities enable us to come near to solving the original problem. A conventional numerical method is then used to solve the resulting system of equations. Theoretical analysis for our proposed strategy is also established, including the convergence theorem and error analysis. The proposed technique’s error bound is confirmed for the test problems as well. The method’s applicability and validity are examined using a variety of instances. The acquired solution is contrasted with other approaches’ solutions described in the literature. This method is better in terms of implementation, adaptability and computing efficiency for solving other partial differential equations as a result of the comparison of the proposed method to existing methods used to solve the fractional diffusion-wave equation.
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