Abstract

In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.

Highlights

  • Differential equations (DEs) can be resolved by a diversity of procedures, analytical and numerical

  • There are numerous analytic methods for verdict on the results of DEs; there occur quite a numeral of DEs that cannot be explained analytically. is means that the result cannot be articulated as a summation of a fixed numeral of basic functions

  • Numerical method is based on the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) and the collocation method [13]

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Summary

Introduction

Differential equations (DEs) can be resolved by a diversity of procedures, analytical and numerical. Numerous procedures have been offered to resolve fractional order DEs comprising the Bernstein wavelets method [1], Shehu variational iteration method [2], Chebyshev spectral collocation approach [3], Taylor wavelet technique [4], operational matrix approach [5], fractional natural decomposition method [6]. E general form of the fractional order CRDE is as follows [19]: zπΦ(χ, Υ) z2Φ(χ, Υ). Verdict on the results of fractional order CRDEs is a fascinating zone for the researchers. It is a very useful method to resolve the entire natures of DEs. Elzaki transform was defined for functions of exponential order. We build Elzaki residual power series solutions for CRDEs. Further, few problems are solved to illustrate the capability, the potentiality, and the simplicity of the proposed method.

Some New Results
Demonstrating the ERPSM for the CRDEs
Approximate and Closed Form Solutions of Nonlinear CRDEs
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