Fifth order, quasi-linear, non-constant separant evolution equations are of the form u t = A ∂ 5 u ∂x 5 +B, where A andB are functions of x, t, u and of the derivatives of u with respect to x up to order 4. We use the existence of a "formal symmetry", hence the existence of "canonical conservation laws" ρ (i) , i = −1, . . . , 5 as an integrability test. We define an evolution equation to be of the KdV-Type, if all odd numbered canonical conserved densities are nontrivial. We prove that fifth order, quasi-linear, non-constant separant evolution equations of KdV type are polynomial in the function a = A 1/5 ; a = (αu 2 3 + βu 3 + γ) −1/2 , where α, β and γ are functions of x, t, u and of the derivatives of u with respect to x up to order 2. We determine the u 2 dependency of a in terms of P = 4αγ − β 2 > 0 and we give an explicit solution, showing that there are integrable fifth order non-polynomial evolution equations.

The study in this paper mainly concerns the inverse problem of determining an unknown source function in the linear fractional differential equation with variable coefficient using Adomian decomposition method (ADM). We apply ADM to determine the continuous right hand side functionsfxandftin the heat-like diffusion equationsDtαux,t=hxuxxx,t+fxandDtαux,t=hxuxxx,t+ft, respectively. The results reveal that ADM is very effective and simple for the inverse problem of determining the source function.

We develop a high-order fixed point type method to approximate a multiple root. By using three functional evaluations per full cycle, a new class of fourth-order methods for this purpose is suggested and established. The methods from the class require the knowledge of the multiplicity. We also present a method in the absence of multiplicity for nonlinear equations. In order to attest the efficiency of the obtained methods, we employ numerical comparisons alongside obtaining basins of attraction to compare them in the complex plane according to their convergence speed and chaotic behavior.

Fifth order, quasi-linear, non-constant separant evolution equations are of the form u t = A ∂ 5 u ∂x 5 + B, where A and B are functions of x, t, u and of the derivatives of u with respect to x up to order 4. We use the existence of a "formal symmetry", hence the existence of "canonical conservation laws" ρ (i) , i = −1, . . . , 5 as an integrability test. We define an evolution equation to be of the KdV-Type, if all odd numbered canonical conserved densities are nontrivial. We prove that fifth order, quasi-linear, non-constant separant evolution equations of KdV type are polynomial in the function a = A 1/5 ; a = (αu 2 3 + βu 3 + γ) −1/2 , where α, β and γ are functions of x, t, u and of the derivatives of u with respect to x up to order 2. We determine the u 2 dependency of a in terms of P = 4αγ − β 2 > 0 and we give an explicit solution, showing that there are integrable fifth order non-polynomial evolution equations.