Numerical Simulations for the Space-Time Variable Order Nonlinear Fractional Wave Equation

Abstract

The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered. The concept of variable order fractional derivative is considered in the sense of Caputo. The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presented.

1. Introduction

It is well known that the fractional calculus definitions are extensions of the usual calculus definitions [1–8], where the orders need not to be positive integers. On the other hand, the variable order calculus is a natural extension of the constant order (integer or fractional) calculus. In this sense, the order may function in any variable such as time and space variables or a system of other variables [9, 10]. In general, one can say that this extension is introduced by Samko and Ross in [11], where Marchaud fractional derivative and Riemann-Liouville derivative are extended to the variable order cases the order in this case is a function in the space variable only. Many authors have introduced different definitions of variable order differential operators, each of these with a specific meaning to suit desired goals. These definitions such as Riemann-Liouville, Grünwald, Caputo, Riesz [3, 12–16], and some notes as Coimbra definition [17, 18].

Coimbra in [17] used Laplace transform of Caputo’s definition of the fractional derivative as the starting point to suggest a novel definition for the variable order differential operator. Because of its meaningful physical interpretation, Coimbra’s definition is better suited for modeling physical problems. The variable order differentials are an important tool to study some systems such as the control of nonlinear viscoelasticity oscillator (for more details see [17–19] and the references cited therein), where the order changes with respect to a parameter or more parameters.

In the following, we present the basic definition for the variable order fractional derivatives which we will use in this paper.

Definition 1 (see [14]). The Caputo space variable order derivative is defined as follows: 𝐷_𝑥^{∝(𝑥,𝑡)}𝑢()=1𝑥,𝑡Γ((×𝑛−∝𝑥,𝑡))_𝑥01(𝑥−𝜉)^{∝(𝑥,𝑡)−𝑛+1}𝜕^𝑛𝑢(𝜉,𝑡)𝜕𝜉^𝑛𝑑𝜉, (1) where 0<∝(𝑥,𝑡)<1.

The main aim of this work is to use the explicit finite difference method (EFDM) to study numerically the following nonlinear space-time variable order wave equation: 𝜕^{𝛽(𝑥,𝑡)}𝑢(𝑥,𝜏)𝜕𝜏^{𝛽(𝑥,𝑡)}()𝜕=𝐵𝑥,𝑡^{∝(𝑥,𝑡)}𝑢(𝑥,𝜏)𝜕𝑥^{∝(𝑥,𝑡)}(),()()+𝑓𝑢,𝑥,𝑡1<∝𝑥,𝑡,𝛽𝑥,𝑡≤2, (2) subject to initial conditions 𝑢(𝑥,0)=𝜑₁(𝑥),𝑢_𝑡(𝑥,0)=𝜑₂(𝑥), (3) and the following boundary conditions 𝑢(0,𝑡)=Ψ₁(𝑥),𝑢(𝑎,𝑡)=Ψ₂(𝑥), (4) where 0≤𝑥<𝑎,0≤𝑡<𝑇,𝐵(𝑥,𝑡)>0 is a constant, Ψ₁(𝑥),Ψ₂(𝑥) are smooth functions, and 𝑓(𝑢,𝑥,𝑡) is a nonlinear scour term that satisfies the Lipschitz condition, that is, 𝑓𝑢₁,𝑥,𝑡−𝑓𝑢₂,𝑥,𝑡≤𝐿𝑢₁−𝑢₂, (5) where the constant 𝐿>0 is called a Lipschitz constant for 𝑓.

2. Discretization for EFDM

In this section, EFDM is used to study the model problem (2), then the space-time solutions domain will be discretized. The discrete form for the pervious Caputo derivative can be written as follows: 𝐷_𝑥^{∝(𝑥,𝑡)}𝑢()=1𝑥,𝑡Γ((×2−∝𝑥,𝑡))_𝑥01(𝑥−𝜉)^{∝(𝑥,𝑡)−2+1}𝜕²𝑢(𝜉,𝑡)𝜕𝜉²=1𝑑𝜉Γ(2−∝(𝑥,𝑡))^𝑖−1_𝑘=0^(𝑘+1)ℎ_𝑘ℎ𝑧^{1−∝(𝑥,𝑡)}×𝜕²𝑢(𝑥−𝑧,𝜏)𝜕𝑧²≈1𝑑𝑧,𝑧=𝑥−𝜉Γ((×2−∝𝑥,𝑡))_{𝑖−1𝑘=0}(𝑢(())()(()ℎ𝑥−𝑘−1ℎ,𝑡−2𝑢𝑥−𝑘ℎ,𝑡+𝑢𝑥−𝑘+1ℎ,𝑡))²⁻¹×^(𝑘+1)ℎ_𝑘ℎ𝑧^1−∝𝑑𝑧. (6) Then, 𝐷_𝑥^{∝(𝑥,𝑡)}𝑢()≈ℎ𝑥,𝑡_{2−∝(𝑥,𝑡)}Γ((×3−∝𝑥,𝑡))_{𝑖−1𝑘=0}(𝑢(())()(()ℎ𝑥−𝑘−1ℎ,𝑡−2𝑢𝑥−𝑘ℎ,𝑡+𝑢𝑥−𝑘+1ℎ,𝑡))²⁻¹×(𝑘+1)^{2−∝(𝑥,𝑡)}−𝑘^{2−∝(𝑥,𝑡)}. (7) Now, pick two positive integers 𝑁, 𝑀 and define the step size of space and time by ℎ, 𝜏, respectively, where ℎ=𝑎/𝑀 and 𝜏=𝑇/𝑁. Also we introduce the following notations: 𝑥_𝑖𝑡=𝑖ℎ,for𝑖=1,2,…,𝑁,_𝑗=𝑗𝜏,for𝑗=1,…,𝑀, (8)𝑢^𝑗_𝑖≈𝑢(𝑥_𝑖,𝑡_𝑗), 𝐵^𝑗_𝑖=𝐵(𝑥_𝑖,𝑡_𝑗), and 𝑓^𝑗_𝑖=𝑓(𝑢^𝑗_𝑖,𝑥_𝑖,𝑡_𝑗). Then, 𝐷_𝑥^{∝(𝑥,𝑡)}𝑢()=ℎ𝑥,𝑡_{−𝛼𝑗𝑖}Γ3−𝛼^𝑗_𝑖^𝑖−1_𝑘=0𝑢^𝑗_{𝑖−𝑘+1}−2𝑢^𝑗_𝑖−𝑘+𝑢^𝑗_{𝑖−𝑘−1}×(𝑘+1)^{2−𝛼𝑗𝑖}−𝑘^{2−𝛼𝑗𝑖}. (9) By the same way, we have 𝐷_𝑡^{𝛽(𝑥,𝑡)}𝑢()=𝜏𝑥,𝑡_{−𝛽𝑗𝑖}Γ3−𝛽^𝑗_𝑖^𝑗−1_𝑘=0𝑢_𝑖^{𝑗−𝑘+1}−2𝑢_𝑖^𝑗−𝑘+𝑢_𝑖^{𝑗−𝑘−1}×(𝑘+1)^{2−𝛽𝑗𝑖}−𝑘^{2−𝛽𝑗𝑖}. (10) For simplicity, let us define 𝑅^𝑗_𝑖=𝐵^𝑗_𝑖ℎ^{−∝𝑗𝑖}Γ3−∝^𝑗_𝑖,𝑄^𝑗_𝑖=Γ3−𝛽^𝑗_𝑖𝜏^{−𝛽𝑗𝑖},𝐺^𝑗_𝑘=(𝑘+1)^{2−∝𝑗𝑖}−𝑘^{2−∝𝑗𝑖},𝐻^𝑘_𝑖=(𝑘+1)^{2−𝛽𝑗𝑖}−𝑘^{2−𝛽𝑗𝑖}. (11) Then, we can rewrite (2) in the following form: ^𝑗−1_𝑘=0𝑢_𝑖^{𝑗−𝑘+1}−2𝑢_𝑖^𝑗−𝑘+𝑢_𝑖^{𝑗−𝑘−1}𝐻^𝑘_𝑖≈𝑄^𝑗_𝑖𝑅^𝑗_𝑖^𝑖−1_𝑘=0𝑢^𝑗_{𝑖−𝑘+1}−2𝑢^𝑗_𝑖−𝑘+𝑢^𝑗_{𝑖−𝑘−1}𝐺^𝑗_𝑘+𝑄^𝑗_𝑖𝑓^𝑗_𝑖, (12) that is, 𝑢_𝑖^𝑗+1=2𝑢^𝑗_𝑖−𝑢_𝑖^𝑗−1−^𝑗−1_𝑘=1𝑢_𝑖^{𝑗−𝑘+1}−2𝑢_𝑖^𝑗−𝑘+𝑢_𝑖^{𝑗−𝑘−1}𝐻^𝑘_𝑖+𝑄^𝑗_𝑖𝑅^𝑗_𝑖^𝑖−1_𝑘=0𝑢^𝑗_{𝑖−𝑘+1}−2𝑢^𝑗_𝑖−𝑘+𝑢^𝑗_{𝑖−𝑘−1}𝐺^𝑗_𝑘+𝑄^𝑗_𝑖𝑓^𝑗_𝑖,𝑢_𝑖^𝑗+1=2−𝐻¹_𝑖𝑢^𝑗_𝑖−^𝑀−2_𝑘=2𝐻_𝑖^𝑘−2−2𝐻_𝑖^𝑘−1+𝐻^𝑘_𝑖𝑢_𝑖^{𝑗−𝑘+1}−𝐻_𝑖^𝑗−2−2𝐻_𝑖^𝑗−1𝑢¹_𝑖−𝐻_𝑖^𝑗−1𝑢⁰_𝑖+𝑄^𝑗_𝑖𝑅^𝑗_𝑖^𝑖−1_𝑘=0𝑢^𝑗_{𝑖−𝑘+1}−2𝑢^𝑗_𝑖−𝑘+𝑢^𝑗_{𝑖−𝑘−1}𝐺^𝑗_𝑘+𝑄^𝑗_𝑖𝑓^𝑗_𝑖. (13) The previous equation can be expressed in the following matrix form: 𝑈⁰_𝑖=∅₁,𝑈¹_𝑖=𝑈⁰_𝑖+𝜏∅₂, (14) and for 𝑗≥2𝑈_𝑖^𝑗+1=𝐴^𝑗𝑈^𝑗_𝑖−^𝑀−2_𝑘=2𝐻_𝑖^𝑘−2−2𝐻_𝑖^𝑘−1+𝐻^𝑘_𝑖𝑈_𝑖^{𝑗−𝑘+1}−𝐻_𝑖^𝑗−2−2𝐻_𝑖^𝑗−1𝑈¹_𝑖−𝐻_𝑖^𝑗−1𝑈⁰_𝑖+𝐹^𝑗, (15) where 𝐹^𝑗=(𝑄^𝑗_𝑖𝑓(𝑢^𝑗_𝑚−1,𝑥_𝑚−1,𝑡_𝑗),…,𝑄^𝑗_𝑖𝑓(𝑢^𝑗₁,𝑥₁,𝑡_𝑗))^𝑇,𝑈^𝑗=(𝑢^𝑗_𝑀−1,𝑢^𝑗_𝑀−2,…,𝑢^𝑗₁)^𝑇, ∅₁=𝜑₁𝑥₁,𝜑₁𝑥₂,…,𝜑₁𝑥_𝑁^𝑇,∅₂=𝜑₂𝑥₁,𝜑₂𝑥₂,…,𝜑₂𝑥_𝑁^𝑇, (16) and 𝐴^𝑗=(𝑎^𝑗_𝑛𝑚) is a matrix with the following coefficients: 𝑎^𝑗_𝑛𝑚=𝑄^𝑗_𝑛𝑅^𝑗_𝑛𝐺^𝑗_𝑛−1𝑄,where𝑚=1,_𝑗𝑛𝑅^𝑗_𝑛𝐺^𝑗_𝑛−𝑚−2𝐺^𝑗_{𝑛−𝑚+1}+𝜃𝐺^𝑗_{𝑛−𝑚+2},where𝑚≤𝑛,2−𝐻^𝑗_𝑛+𝑄^𝑗_𝑛𝑅^𝑗_𝑛𝜃𝐺^𝑗₁−2𝐺^𝑗₀𝑄,where𝑚=𝑛+1,_𝑗𝑛𝑅^𝑗_𝑛𝐺^𝑗₀,where𝑚=𝑛+2,0,where𝑚>𝑛+2,𝜃=0,where𝑚=2,1,otherwise, (17) for 𝑛=1,2,…,𝐾−1, and 𝑚=1,2,…,𝐾−1. Also, we note that ‖𝐴‖_∞=max^𝐾_{1≤𝑛≤𝐾𝑚=1}𝑎_𝑛𝑚=max_{1≤𝑛≤𝐾}2−𝐻^𝑛_𝑖=2−𝐻⁰_𝑖, (18) then ‖𝐴‖_∞=1.

Lemma 2. The coefficients 𝐺^𝑗_𝑘 and 𝐻^𝑘_𝑖 satisfy the following conditions:(1)𝐺^𝑗₀=1, and 𝐻⁰_𝑖=1,(2)𝐺^𝑗_𝑘>𝐺^𝑗_𝑘+1, and 𝐻^𝑘_𝑖>𝐻_𝑖^𝑘+1, for 𝑘=0,1,…

3. The Stability Analysis and the Truncation Error

Let us consider 𝑊^𝑗+1 and 𝑈^𝑗+1 to be two different numerical solutions of (15) with initial values given by 𝑊⁰ and 𝑈⁰, respectively.

Theorem 3. The explicit method approximation defined by (15) to the variable order space-time wave equation (2) is unconditionally stable, that is, 𝑊^𝑗+1−𝑈^𝑗+1≤𝐶𝑊⁰−𝑈⁰,𝑓𝑜𝑟𝑎𝑛𝑦𝑗. (19)

Proof. Let us define 𝑊^𝑗+1−𝑈^𝑗+1=𝜀^𝑗+1.
From (15) we have 𝜀_𝑖^𝑗+1=𝐴^𝑗𝜀^𝑗_𝑖−^𝑀−2_𝑘=2𝐻_𝑖^𝑘−2−2𝐻_𝑖^𝑘−1+𝐻^𝑘_𝑖𝜀_𝑖^{𝑗−𝑘+1}−𝐻_𝑖^𝑗−2−2𝐻_𝑖^𝑗−1𝜀¹_𝑖−𝐻_𝑖^𝑗−1𝜀⁰_𝑖+𝐹^𝑗_𝜀, (20) where 𝐹^𝑗_𝜀=𝑄^𝑗_𝑚−1𝑓𝑢^𝑗_𝑚−1,𝑥_𝑚−1,𝑡_𝑗−𝑄^𝑗_𝑚−1𝑓𝑤^𝑗_𝑚−1,𝑥_𝑚−1,𝑡_𝑗,…,𝑄^𝑗₁𝑓𝑢^𝑗₁,𝑥₁,𝑡_𝑗−𝑄^𝑗₁𝑓𝑤^𝑗₁,𝑥₁,𝑡_𝑗^𝑇≤𝑄^𝑗_𝑚−1𝐿^𝑗_𝑚−1𝜀^𝑗_𝑚−1,…,𝑄^𝑗₁𝐿^𝑗₁𝜀^𝑗₁^𝑇=Δ𝐹^𝑗𝜀^𝑗, (21) and Δ𝐹^𝑗=diag(𝑄^𝑗_𝑚−1𝐿^𝑗_𝑚−1,…,𝑄^𝑗₁𝐿^𝑗₁)^𝑇.
Noting that |𝐿^𝑗_𝑖|≤𝐿, for any 𝑖,𝑗.
Let 𝑄=max{𝑄^𝑗_𝑚−1,…,𝑄^𝑗₁}. From (20), we have ‖𝐴^𝑗+Δ𝐹^𝑗‖_𝑚≤(2+𝑄𝐿), then 𝜀_𝑖^𝑗+1_∞≤𝐴^𝑗+Δ𝐹^𝑗_∞𝜀^𝑗_𝑖∞+^𝑀−2_𝑘=2𝐻_𝑖^𝑘−2−2𝐻_𝑖^𝑘−1+𝐻^𝑘_𝑖𝜀_𝑖^{𝑗−𝑘+1}_∞+𝐻_𝑖^𝑗−2−2𝐻_𝑖^𝑗−1𝜀¹_𝑖∞+𝐻_𝑖^𝑗−1𝜀⁰_𝑖∞. (22) Now, we analyze the stability via mathematical induction [10]. From (14) we have ‖𝜀¹_𝑖‖_∞≤𝐶‖𝜀⁰_𝑖‖_∞, where 𝐶 is a constant.
Now, assume that ‖𝜀^𝑗_𝑖‖_∞≤𝐶‖𝜀⁰_𝑖‖_∞, then from (22), we have 𝜀_𝑖^𝑗+1_∞≤𝐶2+𝑄𝐿𝜀⁰_𝑖∞+^𝑀−2_𝑘=2𝐻_𝑖^𝑘−2−2𝐻_𝑖^𝑘−1+𝐻^𝑘_𝑖𝐶𝜀⁰_𝑖∞+𝐶𝐻_𝑖^𝑗−2−2𝐻_𝑖^𝑗−1𝜀⁰_𝑖∞+𝐻_𝑖^𝑗−1𝜀⁰_𝑖∞≤𝐶₁𝜀⁰_𝑖∞. (23) Then, the theorem holds.

Lemma 4. Let ₀𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗=ℎ^{−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗×^𝑗−1_𝑘=0𝐺^𝑗_𝑖𝑢^𝑗_{𝑖−𝑘+1}−2𝑢^𝑗_𝑖−𝑘+𝑢^𝑗_{𝑖−𝑘+1} (24) be a smooth function; then 𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗−𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗=𝑂(ℎ). (25)

Proof. In terms of standard centered difference formula, we have ₀𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗=ℎ^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0𝐺^𝑗_𝑖𝜕²𝑢𝑥−𝑗ℎ,𝑡𝜕𝑧²+𝑂ℎ²=ℎ^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0𝐺^𝑗_𝑖𝜕²𝑢𝑥−𝑗ℎ,𝑡𝜕𝑧²+ℎ^{2−∝(𝑥𝑖,𝑡𝑗)}𝑘^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗𝑂ℎ²=ℎ^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0𝐺^𝑗_𝑖𝜕²𝑢𝑥−𝑗ℎ,𝑡𝜕𝑧²+𝑥^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗𝑂ℎ²=ℎ^{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0𝐺^𝑗_𝑖𝜕²𝑢𝑥−𝑗ℎ,𝑡𝜕𝑧²+𝑂ℎ². (26) By the integral mean value theorem, we have ₀𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗=1Γ2−∝𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0^(𝑗+1)ℎ_𝑗ℎ𝑧^{1−∝(𝑥𝑖,𝑡𝑗)}𝜕²𝑢(𝑥−𝑧,𝑡)𝜕𝑧²=ℎ𝑑𝑧_{2−∝(𝑥𝑖,𝑡𝑗)}Γ3−∝𝑥_𝑖,𝑡_𝑗, (27) where 𝜉_𝑗∈[𝑗ℎ,(𝑗+1)ℎ]. Combining the pervious two formulae, we have 𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗−𝐷^{∝(𝑥𝑖,𝑡𝑗)}_𝑥𝑢𝑥_𝑖,𝑡_𝑗=ℎ^{2−𝛼(𝑥𝑖,𝑡𝑗)}Γ3−𝛼𝑥_𝑖,𝑡_𝑗×^𝑘−1_𝑗=0𝐺^𝑗_𝑖𝜕²𝑢𝑥−𝑗ℎ,𝑡𝜕𝑧²−𝜕²𝑢𝑥−𝜉_𝑗,𝑡𝜕𝑧²+𝑂ℎ²=ℎ^{2−𝛼(𝑥𝑖,𝑡𝑗)}Γ3−𝛼𝑥_𝑖,𝑡_𝑗^𝑘−1_𝑗=0𝐺^𝑗_𝑖⋅𝑂(ℎ)+𝑂ℎ²=ℎ^{2−𝛼(𝑥𝑖,𝑡𝑗)}𝑘^{2−𝛼(𝑥𝑖,𝑡𝑗)}Γ3−𝛼𝑥_𝑖,𝑡_𝑗⋅𝑂(ℎ)+𝑂ℎ²=𝑂(ℎ)+𝑂ℎ²=𝑂(ℎ). (28) Now, by using Lemma 4, we can derive the truncation error of explicit finite difference scheme (14). It has a local truncation error of 𝑂(𝜏) (from the left side) and 𝑂(ℎ) (from the right side).

Remark 5. The pervious explicit method was shown to be stable. This method is consistent with a local truncation error which is 𝑂(𝜏)+𝑂(ℎ). Therefore, according to the Lax Equivalence Theorem [2], it converges at this rate.

4. Numerical Examples

Example 1. Consider the following variable-order linear fractional wave equation: 𝜕^{𝛽(𝑥,𝑡)}𝑢(𝑥,𝑡)𝜕𝜏^{𝛽(𝑥,𝑡)}∝()𝜋=−0.5cos𝑥,𝑡2×𝜕^{∝(𝑥,𝑡)}𝑢(𝑥,𝑡)𝜕𝑥^{∝(𝑥,𝑡)}+𝑓(𝑢,𝑥,𝑡), (29) with ∝(𝑥,𝑡)=1.5+0.5𝑒^{−(𝑥𝑡)2−1},𝛽(𝑥,𝑡)=1.5+0.25cos(𝑥)sin(2𝑡), and 𝑓(𝑢,𝑥,𝑡)=2𝑢𝑡²−𝑡+1²×+116𝑥^{2−∝(𝑥,𝑡)}Γ((+3−∝𝑥,𝑡))6𝑥^{3−∝(𝑥,𝑡)}Γ((,4−∝𝑥,𝑡)) (30) subjected to the following initial conditions: 𝑢(𝑥,0)=𝜑(𝑥)=𝑥²(),𝑢8−𝑥_𝑡(𝑥,0)=Ψ(𝑥)=0, (31) where 𝑋_𝑎=0,  𝑋_𝑏=8, and 𝑇=1.

The exact solution of this problem when 𝛽=2 is 𝑢(𝑥,𝑡)=𝑥²(8−𝑥)(𝑡²+1).

In Figure 1, a comparison between the numerical and the exact solutions when 𝛽=2 at 𝑡=0.416 is presented.

Figure 1 

In Figures 2(a) and 2(b), we report the approximate solutions at 𝑡=0.052 and 𝑡=0.78, respectively.

(a)

(b)

Figure 2 

In Figures 3, 4, and 5, respectively, we report the approximate solutions in three dimensions, where the axis’s are (𝑡,𝑥,𝑢),(alfa,𝑥,𝑢), and (beta,𝑥,𝑢), respectively.

Figure 3 

Figure 4 

Figure 5 

Example 2. Consider the following variable-order nonlinear fractional wave equation: 𝜕^{𝛽(𝑥,𝑡)}𝑢(𝑥,𝑡)𝜕𝑡^{𝛽(𝑥,𝑡)}𝜕=2cos𝑡^{∝(𝑥,𝑡)}𝑢(𝑥,𝑡)𝜕𝑥^{∝(𝑥,𝑡)}+𝑓(𝑢,𝑥,𝑡), (32) with ∝(𝑥,𝑡)=2−cos²(𝑥)sin²(𝑡),𝛽(𝑥,𝑡)=1.8+0.5𝑒^{−(𝑥𝑡)2−1} and 𝑢(𝑥,0)=𝜑(𝑥)=1+sin𝑥,𝑢_𝑡(𝑥,0)=Ψ(𝑥)=0, where 0≤𝑥≤10,𝑇=1, and 𝑓(𝑢,𝑥,𝑡)=𝑢²−sin²(𝑥)−cos²(𝑡).
This problem has the following exact solution, when ∝=2𝑢(𝑥,𝑡)=sin𝑥+cos𝑡. (33)

In Figure 6(a), we report the numerical solution when ∝(𝑥,𝑡), 𝛽(𝑥,𝑡) are variables at 𝑡=0.52 and the exact solutions when ∝=𝛽=2.

(a)

(b)

(c)

Figure 6 

In Figures 6(b) and 6(c), we report the approximate solution at 𝑡=0.052 and 𝑡=0.78, respectively.

Figures 7, 8, and 9 show the approximate solution in three dimensions, where the axes are (𝑡,𝑥,𝑢), (alfa,𝑥,𝑢) and (beta,𝑥,𝑢), respectively.

Figure 7 

Figure 8 

Figure 9 

5. Conclusions

In this paper, numerical studies using a simple explicit FDM for solving the variable order space-time wave equation are presented. The stability analysis and the truncation error of the proposed method are proved. Some test examples are given, and the results obtained by the method are compared with the exact solutions in integer order cases. Several figures are presented to simulate the solutions behaviors when the variable orders change with respect to space and time. The comparison certifies that FDM gives good results. Summarizing these results, we can say that the finite difference method in its general form gives reasonable calculations, easy to use, and can be applied for the variable order differential equations in general form. All results were obtained by using MATLAB version 7.6.0 (R2008a).