Abstract
Present paper proposes a new technique to solve uncertain beam equation using double parametric form of fuzzy numbers. Uncertainties appearing in the initial conditions are taken in terms of triangular fuzzy number. Using the single parametric form, the fuzzy beam equation is converted first to an interval-based fuzzy differential equation. Next, this differential equation is transformed to crisp form by applying double parametric form of fuzzy number. Finally, the same is solved by homotopy perturbation method (HPM) to obtain the uncertain responses subject to unit step and impulse loads. Obtained results are depicted in terms of plots to show the efficiency and powerfulness of the methodology.
Highlights
Theory of differential equations plays a vital role to model physical and engineering problems such as in solid and fluid mechanics, viscoelasticity, biology, physics, and other areas of science
To overcome the uncertainty and vagueness, one may use fuzzy environment in parameters, variables, and initial condition in place of crisp ones. With these uncertainties the general differential equations turn into fuzzy differential equations (FDEs)
We present the numerical solution of uncertain beam equation using homotopy perturbation method (HPM)
Summary
Theory of differential equations plays a vital role to model physical and engineering problems such as in solid and fluid mechanics, viscoelasticity, biology, physics, and other areas of science. To overcome the uncertainty and vagueness, one may use fuzzy environment in parameters, variables, and initial condition in place of crisp (fixed) ones. With these uncertainties the general differential equations turn into fuzzy differential equations (FDEs). Various numerical methods for solving fuzzy differential equations are introduced in [6,7,8,9,10,11,12,13,14,15,16]. Variational iteration method is discussed by Allahviranloo et al [11] to obtain the exact solutions of fuzzy wave-like equations with variable coefficients. The concept of generalized H-differentiability is studied by Chalco-Cano and Roman-Flores [12] to solve fuzzy differential equations. Akin et al [15] developed an algorithm based on αcut of a fuzzy set for the solution of second-order fuzzy initial
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